Mixing protocols determine liquid–liquid phase separation dynamics in polyelectrolyte complex coacervation

Abstract

Polyelectrolyte complex coacervation underpins many critical biological processes, yet how different initial mixing protocols determine its liquid-liquid phase separation (LLPS) dynamics remains unclear. Using molecular dynamics simulations, we show that when polycations and polyanions are initially randomly mixed, coacervate domain growth exhibits transient t^1/2 scaling, driven by polymer network formation. This phase is followed by either t¹ scaling due to hydrodynamic pumping or t^1/3 scaling from droplet coarsening, depending on the initial mixing degree. Conversely, starting with spatially separated domains of polycations and polyanions-mimicking LLPS in certain marine organisms-leads to rapid coacervate formation, with early-stage growth following distinct t^2/3 scaling due to strong electrostatic attraction, followed by continued growth via polymer accumulation. Both protocols yield significantly faster dynamics than systems initialized with preformed polyion pairs, which exhibit classical t^1/3 scaling characteristic of droplet coarsening. These findings highlight the profound impact of initial conditions on LLPS dynamics in polyelectrolyte systems.

Liquid-liquid phase separation is a process of interest, but the dynamics are still not fully understood. Here, the authors use molecular dynamics to consider the role of mixing protocol in polyelectrolyte complex coacervation.

Introduction

Polyelectrolyte complex coacervation refers to the liquid-liquid phase separation (LLPS) driven by electrostatic association of oppositely charged polyelectrolytes in an aqueous environment, resulting in the formation of a polymer-rich coacervate phase coexisting with a dilute phase^1,2. In recent years, this LLPS process has received significant attention due to its relevance to various critical biological phenomena, including the formation of membraneless organelles^3–6, the progression of age-related diseases^7,8, and underwater adhesion of marine organisms^9–11. While substantial advances have been made in understanding the thermodynamics of LLPS over the past decade, the dynamics of LLPS — including the kinetic pathways and rates of phase separation — remains relatively underexplored. LLPS dynamics play a crucial role in the spatiotemporal regulation of intracellular organelles¹², and can significantly impact the life survival of marine species in harsh undersea conditions^9–11.

Recently, we studied LLPS dynamics in polyelectrolyte complex coacervation starting from an initial state of preformed polyion pairs¹³. For symmetric mixtures, the growth of the coacervate domains was shown to follow the classical droplet coarsening mechanism¹⁴, characterized by a power-law growth of the coacervate domain as R ~ t^β, with β = 1/3. We shall refer to this pathway as the “thermodynamic pathway” on consideration that, thermodynamically, coacervation can be considered a two-step process: the formation of polyion pairs, and the condensation of polyion pairs into a macroscopic coacervate phase^15–18. Other factors, such as polyelectrolyte charge asymmetry^13,19, intracellular activities^20,21, and coupling to elastic deformation of the cellular matrix^22–24 can further suppress the LLPS dynamics. Consequently, the thermodynamic pathway often results in relatively slow coacervation dynamics.

While the thermodynamic pathway can describe the relatively slow coarsening dynamics of coacervate droplets starting from dilute solutions, in reality, LLPS often starts with dense polyelectrolyte (PE) mixtures and exhibits significantly faster phase separation dynamics. For example, many experiments trigger LLPS by applying environmental cues such as lights²² and pH²⁵ to concentrated, well-mixed polyelectrolyte solutions, which results in complete phase separation in a few minutes^22,25,26. This relatively rapid phase separation can be conceptually understood by the facilitated contacts between oppositely charged polymers during mixing^27–29, while a quantitative examination of the underlying kinetic pathway and scaling law is not fully explored. Recent work by Yuan and Tanaka³⁰ demonstrated that this pathway involves the formation of a network structure, with a t^1/2 scaling in domain growth, characteristic of viscoelastic phase separation³¹. In marine organisms such as Sandcastle worms and mussels, even faster LLPS dynamics are observed, whereby these organisms produce proteins that rapidly form adhesive coacervates to attach to wet surfaces³². In this scenario, oppositely charged glue proteins are secreted separately into “tubes” (e.g., byssal threads in mussels) at high concentrations, and LLPS is initiated by fluxing these proteins into a shared region, with the coacervates forming in as little as 30 s³². Beyond biological systems, recent studies found that applying shear flux to a mixture of polycation and polyanion solutions can significantly speed up their coarsening, where a clear phase separation is observed on the order of seconds³³. In comparison, without the initial flux, the phase separation of the mixture can only be observed after at least a few minutes^27,34–36. Clearly, different initial mixing protocols can significantly influence the phase separation dynamics of dense polyelectrolyte mixtures. However, the fundamental molecular mechanisms driving the dynamics of LLPS remain poorly understood. In particular, the complete morphological pathways under varying initial mixing protocols, as well as their corresponding time evolution, are not yet fully elucidated.

In this work, using molecular dynamics simulation that explicitly accounts for electrostatics and hydrodynamics, we examine the kinetic pathways and the corresponding LLPS dynamics in polyelectrolyte complex coacervation under several representative initial mixing conditions. We show that different initial mixing protocols lead to unconventional LLPS pathways, which can significantly accelerate the LLPS dynamics, compared to that in the traditional droplet coarsening. These findings highlight the critical role of initial mixing conditions in controlling the LLPS dynamics of complex coacervation, with implications in biotechnological processes, smart materials development, and the design of bioinspired adhesives.

Results

Morphology evolution for distinct initial mixing protocols

We investigate the LLPS dynamics starting from two distinct initial mixing scenarios between the polycations and polyanions. In the first scenario, the system consists of an initially well-mixed solution of polycations and polyanions, mimicking experimental conditions where LLPS starts with a dense mixture and is subsequently triggered by environmental cues^22,25,26. We refer to this protocol as the well-mixed pathway (Fig. 1b). The second scenario involves an initial system composed of spatially segregated domains of polycations and polyanions, mimicking the “secrete and flux" coacervation process observed in certain marine organisms^9–11. We refer to this setup as the flux pathway (Figs. 1c, 3d). For comparison, we include results from the thermodynamic pathway (Fig. 1a).

Fig. 1. Simulation snapshots of the morphology evolution in the thermodynamic pathway, well-mixed pathway, and flux pathway, respectively.

a Thermodynamic pathway: morphology evolution from pre-formed polycation-polyanion pairs via droplet coarsening. The left two panels in (b) and (c) represent early-stage domain growth, and the right two panels show later-stage domain growth. The transition from early stage to later stage is defined as the onset time when the domain growth shows a different exponent β in the power law scaling R ~ t^β. Green and orange beads represent polycation and polyanion monomers, respectively. Water and counterion beads are not shown for clarity.

Fig. 3. Electrochemical potential gradient accelerates the LLPS dynamics in the flux pathway.

a Domain growth at early stage (t₀ ≈ 850τ₀) in the flux pathway. b Domain growth at the later stage (t₁ ≈ 6500τ₀) in the flux pathway. The inset schematically shows the growth mechanism in the later stage. c Spatial distribution of the electrostatic part of the electrochemical potential for the polymer in different simulation times. d Schematic illustration of the relevance of the flux pathway to the coacervation process when mussels produce underwater adhesives. In (a) and (b), the solid line shows the best-fitted scaling relation, and the data points and shaded area, respectively, represent the mean value and standard deviation from 10 individual samples.

We first examine the structural evolution under different pathways, as shown in Fig. 1. In contrast to the thermodynamic pathway (Fig. 1a), where dispersed polycation-polyanion pairs coalesce into progressively larger droplets, the well-mixed and flux pathways exhibit drastically different morphology evolution. For systems starting with a well-mixed solution, the most characteristic feature is the formation of a network structure in the early stages of coacervation, which subsequently breaks down in later stages, eventually evolving into a single coacervate droplet (Fig. 1b). We note that the early-stage network formation and the corresponding dynamics have been recently reported by Yuan and Tanaka³⁰. However, the complete pathway, including the breaking of the network and the later stage dynamics, has not been reported. In systems beginning with spatially segregated polycation and polyanion domains, coacervate droplets form rapidly in the contact zone due to the strong local electrostatic attraction, which is followed by the accumulation of dispersed chains into these interfacial droplets at a later stage, as illustrated in Fig. 1c. The morphological evolution in these two pathways is markedly different from the droplet coarsening mechanism in the thermodynamic pathway (Fig. 1a). Despite the differences in morphological evolution, all pathways ultimately converge to the same final state with a single coacervate droplet¹³. We note that the droplets in our simulations generally have floppy surfaces, indicating they have low surface tension, which is consistent with previous theoretical and experimental studies in coacervate systems^37–39. These results also highlight the efficacy of our GCMe-DPD method in effectively simulating the entire LLPS process of coacervation, capturing the full dynamics from the initial state to the final equilibrium state.

The distinct morphological evolution observed in the well-mixed and flux pathways fundamentally alters the scaling behavior of the domain growth dynamics during LLPS, including both the early and later stages of the process, as detailed below.

Domain growth in the well-mixed pathway

In the early stage of the well-mixed pathway, where the polymer morphology takes on a 3D network structure, the domain growth exhibits a power law scaling of R ~ t^0.46, as shown in Fig. 2a. This growth is significantly faster than that observed under the thermodynamic pathway, where the exponent is 1/3 due to the coarsening of dispersed droplets (Fig. 2d). The rapid network coarsening phenomenon in the early stages of the well-mixed pathway aligns closely with a recent finding by Yuan et. al, where they found polyelectrolyte network structures lead to domain growth scaling as R ~ t^1/230. The connected network formed during this early stage arises from the interplay between local charge balance, low surface tension, and polymer connectivity³⁰. Driven by the minimization of surface energy, this spatial network connectivity and hydrodynamics accelerate the coacervation process, resulting in an exponent close to 1/2, akin to the domain evolution of a viscoelastic polymer network³¹. While the 1/2 scaling appears universal in network-driven phase separation, its theoretical derivation remains elusive, as stated by Tanaka^30,31. Future work is needed to understand this intriguing scaling.

Fig. 2. Domain growth in the well-mixed pathway with varied initial mixing degree and comparison with the thermodynamic pathway.

a–c Domain growth in the well-mixed pathway: (a) Early-stage growth with λ = 16.4 from randomly mixed initial states; (b) Later-stage growth with λ = 16.4; (c) Later-stage growth starting from the ideally mixed initial state (λ = 0). d Droplet coarsening dynamics in the thermodynamic pathway. t₀ is defined as the time when the average domain growth first shows a clear power law^13,22, which corresponds to t₀ = 950τ₀ and 750τ₀ respectively in Figure (a) and (d). The onset of later-stage growth, denoted as t₁, in general has larger variations from sample to sample, ranging from t₁ = 2000τ₀ to t₁ = 6000τ₀. Thus, for each later-stage dynamics, we normalize the time by its own t₁. The solid line shows the best-fitted scaling relation, and the data points and shaded area respectively, represent the mean value and standard deviation from 10 individual samples. e, f Simulation snapshots of the domain growth in the later stage of the well-mixed systems with λ = 16.4 and λ = 0, respectively.

At the later stage of the well-mixed pathway, the network breaks and the domain growth starts to deviate from the t^1/2 scaling. From numerous independent runs, we generally observe two distinct types of later-stage dynamics: one exhibiting R ~ t^1/3 and the other showing R ~ t¹, as illustrated in Fig. 2b, c. The type of growth in the later-stage dependents on the initial degree of mixing of polycations and polyanions. To quantify the initial degree of mixing, we divide the simulation box into N_c small cells of dimensions 10σ₀ × 10σ₀ × 10σ₀, where 10σ₀ is roughly the end-to-end distance of the polyelectrolyte chain in a single-species semi-dilute solution at the same concentration. We then characterize the degree of mixing by the magnitude of macromolecular charge imbalance, defined as

$$\lambda =\frac{1}{N_c}{\sum }_{i=1}^{N_c}\mid n_{p+}\left(i\right)-n_{p-}\left(i\right)\mid ,$$ 1

where n_p±(i) is the number of positive/negative monomers within cell i. On average, each bin contains 100 charged monomers; therefore, λ/2 characterizes the percentage (%) of the deviation from charge neutrality in each bin. Generally, when the initial mixture is prepared by randomly distributing the polyelectrolytes into the simulation box (as detailed in Supplementary Information), the system yields λ ≈ 16.4 for the initial configurations, corresponding to 8.2% deviation from the charge neutrality per bin. Under this condition, the later stage dynamics obeys R ~ t^0.34 (Fig. 2b), resembling the dynamics associated with droplet coarsening. This droplet-coarsening feature is more clearly observed from Supplementary Movie S1 in the Supplementary Information, where small droplets move and coalesce with each other in the later stage until the formation of a single large droplet. Our results show that the domain growth at a later stage is mainly driven by droplet-coarsening instead of the diffusion-limited aggregation (DLA) of solid-like precipitates, consistent with recent simulation results¹³ and recent experimental observation of biomolecular condensate formations^22,23. The coarsening behavior is due to the collapse and fragmentation of the early-stage network caused by the imbalance of local macromolecular charge, resulting in multiple more charge-balanced droplets (Fig. 2e). As such the later-stage coacervation proceeds via droplet coarsening, similar to that in the thermodynamic pathway shown in Fig. 2d. A full domain evolution under this pathway is shown in Supplementary Fig. S4a and movie S1 at the Supplementary Information.

On the other hand, a fully locally charge-balanced initial state can be prepared by carefully positioning polycations and polyanions adjacent to each other, resulting in λ = 0. Under this optimally mixed configuration, the later-stage dynamics exhibit a much faster scaling with R ~ t^0.93(Fig. 2c, f). This scaling is characteristic of the hydrodynamic pumping mechanism¹⁴, where the connectivity of the dense domains is coupled to a strong inertial effect, leading to a scaling transition to R ~ t¹ at the later stage. Such a transition pathway is illustrated in Fig. 2f. In this case, the network formed during the early stage maintains local macromolecular charge balance and does not break into multiple droplets over time. Instead, the network coarsens in a continuous fashion into larger domain sizes, driven by rapid hydrodynamic pumping (Fig. 2f). In rare occasions when the initially mixture is prepared stochastically, we also observe similar hydrodynamic pumping at the later stages when the local macromolecular charges happen to be nearly balanced. A full domain evolution from early stage networking to later-stage hydrodynamic pumping is shown in Supplementary Fig. S4a at the Supplementary Information.

Domain growth in the flux pathway

For the flux pathway initiated with spatially separated domains, the system exhibits very fast LLPS dynamics with R ~ t^0.65 in the early stage, as shown in Fig. 3a. During this initial phase, oppositely charged polyelectrolytes near the interfacial region quickly flow into each other, leading to the rapid formation of a few droplets in the contact region, with many nearby uncomplexed chains, as illustrated in Fig. 1c. The scaling exponent is reminiscent of the hydrodynamic-flux driven coarsening dynamics in deeply quenched binary liquid mixtures, which was predicted to follow R ~ t^2/340. Although the direct connection between these two systems is not clear, both cases involve strong inertia effects. A detailed derivation of this scaling will necessarily couple electrostatics, hydrodynamics, and polymer conformation with the nonequilibrium phase separation, which is well beyond the scope of this work. Here, we provide a phenomenological understanding of how the electrochemical potential gradient in the flux pathway can drive this accelerating LLPS dynamics.

The electrochemical potential consists of a concentration part and an electrostatic part, both favors bringing the two oppositely charged polyelectrolytes into the contact region. The electrostatic potential $\mathrm{\Psi }\left(x,t\right)$ can be approximated by considering a 1-dimensional charge density profile in the x direction, and is given by the Poisson equation:

$$\frac{{\partial }^2\mathrm{\Psi }\left(x,t\right)}{\partial x^2}=-\frac{{\rho }_{ch}\left(x,t\right)}{{ϵ}_0{ϵ}_r},$$ 2

where ${\rho }_{ch}\left(x,t\right)$ is the net charge density distribution (including polycations, polyanions, and the corresponding counterions, see detailed numerical calculation in Supplementary Information). Ψ contributes to the electrochemical potential of the polyelectrolyte as ${\mu }_p^e\left(x\right)=q\left(x\right)\mathrm{\Psi }\left(x\right)$, where $q\left(x\right)$ is the charge density of polyelectrolytes. At the early stage, due to the separation of polycation and polyanion regions, ${\mu }_p^e$ is significantly higher at the two sides than in the interfacial region, as shown in Fig. 3c. Consequently, there is a strong electrochemical potential driving force for the polyions from the two sides to flow into the interfacial region, until $\mathrm{\Delta }{\mu }_p^e$ drops to below 10k_bT (Fig. 3c).

Note that this potential (Donnan potential) is a consequence of macromolecular charge separation and therefore is undiminished by screening. Thus, although the nominal Debye screening length (0.34nm, or 0.48nm if only counterions are considered; see Supplementary Information) is much smaller than our simulation box, this electrochemical potential driving force will take effect. Indeed, the 2/3 law persists much longer when we increase the systems three times larger in the x direction (see details in section E, Supplementary Fig. S5 in Supplementary Information), indicating the time span of this scaling regime is likely to be extended in larger systems. In real systems, the two separated domains are much larger than the box size in our simulation; therefore, we expect the t^2/3 growth to persist until most of the polyions merge with the clusters in the interfacial region. We believe that this flux-driven LLPS is the key for the rapid production of adhesive coacervates in marine organisms such as mussels and Sandcastle worms, as schematically illustrated in Fig. 3d.

At the later stage of the flux pathway, coacervation proceeds by further adsorbing the uncomplexed polyions into the droplets, as well as the coarsening of the droplets towards complete coacervation, as shown in Fig. 1c. Theoretically, droplet coarsening should result in domain growth with R ~ t^1/3. Our results show that the dynamics in the later stage of the flux pathway has scaling R ~ t^0.39, as shown in Fig. 3b. The slightly faster scaling than the expected t^1/3 is probably due to the residual electrochemical driving force for the polyions to migrate towards the interfacial region, as schematically shown in the inset of Fig. 3b, c. A full domain evolution in the flux pathway is given in Supplementary Information, Supplementary Fig. S4b.

The different scaling laws in the different pathways have significant consequences for the macroscopic phase separation time in coacervation. For the flux pathway, if we extrapolate using the early-stage scaling—since we expect this regime to persist for macroscopically large systems—the formation of a coacervate with a domain size of R = 0.6 cm will only take about 10 s (based on taking τ₀ = 7.8 ps and σ₀ = 0.6 nm; see details in Supplementary Information). From the same estimate, in order to reach a state of similar domain size, it will take 47.5 years via the thermodynamic pathway! For the well-mixed pathway, the crossover from the early stage to the later stage is independent of system size¹⁴ and the total time it takes to reach a droplet of macroscopic size is dominated by the later-stage dynamics. With hydrodynamic pumping, macroscopic coacervation can form rapidly due to the linear scaling. If the later stage follows droplet coarsening, the rate of coacervation will be similar to the thermodynamic pathway. Note that such extrapolations are predicated on the assumption that there are no other macroscopic correlation lengths that take effect to change the scaling. The validity of this assumption remains to be tested.

Concentration dependence in early-stage dynamics

The early-stage network formation in the well-mixed pathway depends on polyelectrolyte concentration (ϕ)³⁰. To examine this dependence, we quantify the polyelectrolyte concentration in terms of the overlap concentration ${\varphi }^{*}=0.64\times 6\times N/\left(\pi R_{ee}^3\right)$, where R_ee is the polymer root-mean-square end-to-end distance in a salt-free polyelectrolyte solution⁴¹. The results discussed so far correspond to ϕ = 2.7ϕ^*. As shown in Fig. 4(a), LLPS dynamics are slower in more dilute systems (ϕ = 1.0ϕ^*, 2.0ϕ^*). At ϕ = 2.0ϕ^*, rather than forming a network, oppositely charged polyelectrolytes associate into small droplets, which then coarsen with an exponent of 0.31, resembling spinodal decomposition. At ϕ = 1.0ϕ^*, local associations of polycations and polyanions lead to charge-asymmetric droplets coexisting with naked polyions (Fig. 4b), further slowing coarsening (Supplementary Fig. S6a¹³. On the other hand, if the polymer concentration is significantly lower (ϕ ≪ ϕ^*, but still larger than the coexisting supernatant concentration), the polyions are expected to form charge-balanced pairs first, and the coarsening dynamics should follow R ~ t^1/3. However, direct simulation of LLPS of dilute systems starting with dispersed chains is not computationally feasible within our GCMe-DPD framework, as it would require a much larger system to obtain a statistically reliable domain growth law. In Supplementary Information, we increase the chain length to N = 50 with varying concentrations, the results are qualitatively the same. These findings indicate that network-accelerated LLPS requires relatively high polymer concentrations (ϕ ≈ 3ϕ^*, Fig. S6c), consistent with the recent finding³⁰. However, in real systems, the molecular weights of the polymers are often quite high, naturally satisfying this requirement since ϕ^* ~ N^−2 42. Thus, LLPS dynamics via the formation of a transient network are expected to be more common in experiments.

Fig. 4. Concentration-dependent LLPS dynamics in the early stage of the well-mixed pathway.

a Time evolution of the domain growth in the early stage of the randomly distributed well-mixed system with concentration 1.0ϕ^*, 2.0ϕ^*, and 2.7ϕ^*, with chain length N = 40. b The simulation snapshot for system 1.0ϕ^* at t = 2t₀. In (a), the solid line shows the best-fitted scaling relation, and the data points and shaded area respectively represent the mean value and standard deviation from 10 individual samples.

Discussion

Using molecular dynamics to simulate the complete LLPS process in polyelectrolyte complex coacervation, we highlight how initial mixing protocols give rise to unconventional pathways that can significantly accelerate phase separation dynamics, compared to the droplet coarsening dynamics under the thermodynamic pathway. In systems where polycations and polyanions are initially well-mixed, coacervation is initially sped up by the formation of a transient polymer network, which can be further enhanced by hydrodynamic pumping following the network’s breakdown. In addition, ultrafast coacervation can occur in the flux pathway, driven by electrochemical potential gradients from the initially separated polycation and polyanion domains, mimicking the rapid adhesive processes observed in marine organisms.

Interestingly, the t^2/3 scaling observed in the early stage of the flux pathway persists much longer in a larger system, as shown in Supplementary Fig. S5. We thus believe that in real, larger systems where the separated polymer solution domain is much larger, the t^2/3 scaling will persist to macroscopic length scale and significantly accelerate the coarsening; we refer to this coarsening process as ultrafast LLPS dynamics. In the later stage, the t^0.39 scaling is slightly larger than the classical t^1/3 scaling predicted by the Lifshitz-Slyozov-Wagner (LSW) theory on Ostwald ripening or diffusion-limited growth. This small acceleration is probably due to the residual electrochemical gradient, which provides additional driving force to facilitate the migration of polyions towards the growing domains. It would be highly desirable for future experiments to validate these predictions.

As the focus in this study is using coarse-grained MD simulation to elucidate the scaling behavior in LLPS dynamics of polyelectrolyte complex coacervations on length and time scale well beyond the residue level, our model follows the common practices of coarse-grained simulations^30,43–47 to incorporate some necessary simplifications. For example, our simulation systems treat the excluded volume interaction among all coarse-grained beads (including polyions, counterions, and water) identically, representing a good solvent condition for the polymers at the Flory-Huggins level. As such, our model does not include the solvent quality effects or specific ion effects arising from specific short-range interactions among different species. Besides, our model does not capture the possible water orientational orders and the corresponding induced electrostatic potential at coacervate surfaces^48–50, which could influence the droplet coarsening dynamics. Furthermore, we use a homogeneous dielectric constant to model electrostatics, which does not account for possible local dielectric inhomogeneity arising from polymer aggregations. Accounting for this local dielectric effect requires better theoretical understanding and new methodology development in the construction of coarse-grained models. In principle, all these effects can be captured by all-atom simulations of specific systems. However, such calculations are at present computationally prohibitive for exploring the long-time behavior of large coacervate systems and are well beyond the scope of this work. As this work concerns the scaling behavior in LLPS dynamics of generic coacervate systems on length and time scales well beyond the residue level, we believe that incorporating atomistic details will not qualitatively change our conclusions.

This work provides valuable insight into non-equilibrium LLPS dynamics, revealing how tailored initial conditions can lead to controlled phase separation. Our results can also serve as useful guides for applications in biotechnological processes, smart materials development, and the design of bioinspired adhesives, where rapid and controlled phase transitions are essential.

Methods

Coarse-grained model and simulation methodology

Our simulations employ the Gaussian Core Model with Smeared Electrostatic (GCMe) Interactions^51,52 combined with dissipative-particle-dynamics (DPD) thermostat to capture hydrodynamic interactions^53–55. This method, which we refer to as the Gaussian Core Model with Smeared Electrostatics and Dissipative Particle Dynamics (GCMe-DPD), enables efficient modeling of both the equilibrium and nonequilibrium dynamics of polyelectrolyte solution systems. The soft-core Gaussian potentials used in GCMe can be analytically derived from Gaussian-smeared mass and charges⁵¹. Compared to other coarse-grained models such as the Lennard-Jones based bead-spring models^44,46,47 or other chemically specific models such as MARTINI^56,57, our GCMe-DPD has the advantages in capturing the long-range hydrodynamic and electrostatic interactions, both of which are critical for the non-equilibrium LLPS dynamics. In addition, the high computational efficiency of GCMe-DPD allows us to explore the full LLPS kinetics (from coarsening to complete phase separation) of large coacervate systems, as evidenced in our results in both the main text and Supplementary Information.

Simulation details

In our simulations, the polyelectrolyte monomers, counterions, and solvent molecules (water) are represented as coarse-grained beads, with each bead having a Gaussian distribution in mass and charge density around their center. We set the excluded volume interactions to be the same among all species to represent good solvent quality for the polymers. We use nondimensionalized units in which the mass, energy, and length are scaled by the particle mass m₀, the thermal energy k_BT at room temperature, and the Gaussian-core particle size σ₀, all taken to be unity. Each simulation contains an equal number of fully charged polycations and polyanions of the same chain length, accompanied by their corresponding monovalent mobile counterions to maintain overall charge neutrality. Unless otherwise stated, our simulation box has dimensions of 60σ₀ × 60σ₀ × 60σ₀ and contains 4% polyions with chain length N = 40, 4% counterions, and 92% water, representing a semi-dilute polyelectrolyte solution. The Bjerrum length l_B is set to match the dielectric environment of water at room temperature. The electrostatic strength is l_B/b ≈ 1.37 (where b is the charge spacing along the neighbored polymer bead), which falls into the intermediately charged regime for typical polyelectrolytes. Additional simulation details, including choice of interaction parameters, the simulation setup, and analysis methods, are provided in the Supplementary Information and in our original methodology papers^51,52.

Author contributions

Z.Wu (simulation & theory): Methodology, Software, Investigation, Visualization, Writing - original draft. Z.-G.Wang: Supervision, Writing - review & editing, Funding acquisition. S.Chen: Conceptualization, Visualization, Supervision, Writing - review & editing, Funding acquisition.

Peer review

Peer review information

Nature Communications thanks the anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Data availability

All data is included in the article and/or in the Supplementary Information. Source data were available from the corresponding author upon request.

Code availability

The codes used in this study are available from the corresponding author on request.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-026-68296-5.