Cross-Interaction Softness as a Route to Microphase Separation in Binary Colloidal Systems

TL;DR

The study probes how cross-interaction softness governs phase behavior in binary mixtures of hard and soft particles, using a coarse-grained GEM-3 model, Reference Interaction Site Model (RISM) theory, and molecular dynamics simulations. Four representative systems (HSS, HHS, HSH, SSH) differing only in cross-interaction boundedness are analyzed to isolate the role of cross interactions in macrophase and microphase formation. Penetrable (bounded) cross-interactions are shown to be necessary and sufficient for microphase separation, enabling finite-wavelength compositional order with domain spacing $L=2\\pi/k^*$ even without attractions, while purely hard cross-interactions suppress such ordering. Molecular dynamics reveals hierarchical, multiscale structuring near crossover regimes and highlights qualitative agreement with RISM on phase topology, establishing cross-interaction softness as a design principle for self-assembly in multicomponent colloidal systems.

Abstract

Understanding how interparticle interactions govern phase behavior is central to controlling self-organization in multicomponent soft-matter systems. In particular, the role of cross-interactions between unlike components remains insufficiently understood. Here, we systematically investigate how cross-interaction character controls phase behavior in binary mixtures of hard and soft particles using coarse-grained modeling, Reference Interaction Site Model (RISM) theory, and molecular dynamics simulations. Four representative systems are examined that differ only in whether interactions between unlike particles are bounded or hard-sphere. We show that penetrable (bounded) cross-interactions are both necessary and sufficient to induce microphase separation, even in the absence of attractive forces. Such systems exhibit dispersed states, macrophase separation, and microphase-separated morphologies characterized by finite-wavelength compositional ordering. In contrast, purely hard-sphere cross interactions suppress microphase separation entirely, despite strong local clustering. Comparison between theory and simulations reveals qualitative agreement in phase topology, while simulations additionally capture hierarchical and multiscale ordering near crossover regimes. These findings establish cross-interaction softness as a fundamental design principle for controlling phase behavior in multicomponent colloidal and soft-matter systems.

Figures (11)

• Figure 1: Schematic representation of the coarse-graining approach for a binary polymer mixture, where each polymer is modeled as a single spherical particle.
• Figure 2: Schematic representation of four different mixtures of hard and soft particles. Two mixtures feature cross-interactions governed by soft potentials: (a) Hard--Soft--Soft (HSS) and (b) Hard--Hard--Soft (HHS). The other two mixtures involve cross-interactions described by hard potentials: (c) Hard--Soft--Hard (HSH) and (d) Soft--Soft--Hard (SSH).
• Figure 3: Spinodal phase diagram of the Hard--Soft--Soft (HSS) system showing macrophase separation at low and high fractions of hard colloids and microphase separation at intermediate hard-colloid fractions ($\phi_h$), for various total packing fractions ($\eta$) and interaction strengths ($\varepsilon/k_BT$), as predicted by RISM theory. Representative molecular dynamics snapshots illustrating the corresponding morphologies are also shown. Figure adapted from our previous study by Erigi et al.Erigi_2023.
• Figure 4: (a) Radial distribution functions $g(r)$ for $H_1$--$H_1$ and $H_2$--$H_2$ pairs, (b) radial distribution functions for unlike pairs ($H_1$--$H_2$), (c) structure factors $S_{H_1H_1}(k)$ and $S_{H_2H_2}(k)$ showing the emergence of a finite-$k$ peak with increasing $\eta$, and (d) combined real-space pair correlations at $\eta=0.26$. Long-range oscillations indicate compositional ordering and domain periodicity.
• Figure 5: Phase diagram of the HHS system in the $\eta$--$\phi_{H_2}$ plane obtained from RISM theory. Macrophase separation (MaPS) occurs at extreme compositions for weak cross-interactions, while microphase separation (MiPS) dominates at intermediate $\phi_{H_2}$. Increasing $\varepsilon/k_BT$ stabilizes MiPS and suppresses MaPS.