Hydrodynamic theory of chemically active emulsions

Abstract: We present a systematic theory of chemically active emulsions in the hydrodynamic limit by constructing a thermodynamically consistent framework in which the equilibrium is broken by chemo-stating of fuel molecules. For ternary solutions with active chemical reactions, we obtain an effective dynamics of the conserved field dynamics at long length and time scales. The effective dynamics takes into account the broken time reversal symmetry that manifests itself by the emergence of gradient terms akin to those of Active Model B+, which is a generic theory of active phase separation. In addition to the active coefficients modifying the interfacial energy coefficient, the theory contains higher order terms in the gradient expansion that are necessary to correctly describe the dynamics of chemically active emulsions, extending thus Active Model B+. We study numerically a Flory-Huggins model with active chemical reactions. Our theory predicts the formation of microphases when the effective interfacial energy coefficient becomes negative. Moreover, including noise, we show the existence of bubbly phase separation. We also identify a new type of phase behavior, a dynamic active filament phase. Finally, we discuss the steady state entropy production rate in the system resulting from the active chemical reactions. We observe that the total entropy production rate increases with the driving chemical potential and exhibits a kink-like singularity at the transition to the dynamic active filament phase. Our work shows that the generic behaviors of active phase separation can emerge in chemically active emulsions.

• The paper systematically derives a hydrodynamic model for chemically active emulsions, connecting microscopic chemical kinetics with macroscopic active phase separation.
• It demonstrates that non-equilibrium gradient corrections generate microphase structures, dynamic filaments, and reversed ripening phenomena validated by numerical simulations.
• Measurements of entropy production reveal how chemical driving forces quantify energy dissipation and instabilities in soft, dissipative materials.

Hydrodynamic Theory for Chemically Active Emulsions: Non-equilibrium Phase Behavior and Coarse-grained Modelling

Introduction and Context

Chemically active emulsions, prevalent in soft matter and biological systems, display diverse non-equilibrium phenomena driven by energy-dissipating reactions. The interplay of diffusion, phase separation, and chemical reactions creates a range of behaviors, including microphase formation, arrested coarsening, and dynamic patterns. A unified, thermodynamically consistent coarse-grained theory to describe such emulsions at hydrodynamic scales enables direct connections to the established framework of active matter field theories, notably Active Model B+ (AMB+). This paper presents a detailed derivation of the hydrodynamic limit for ternary mixtures with chemical reactions, systematically reducing the full Cahn-Hilliard-reaction dynamics to a single conserved field with non-equilibrium gradient corrections. Numerical implementation for a Flory-Huggins-type ternary solution directly validates the theoretical predictions.

Model and Theoretical Advances

The study focuses on an incompressible ternary solution consisting of solvent (S) and two solutes (A and B) interconverted by first-order chemical reactions. Critically, the A↔B reactions are mediated by both passive (equilibrium, detailed balance-satisfying) and active (fuel-consuming, driven out of equilibrium) pathways. The active reaction is controlled by a chemostatted fuel, e.g., ATP, characterized by a driving chemical potential Δμ.

Two collective fields are identified: the total solute density, $\psi = \phi_A + \phi_B$, is conserved and slow, while the reaction extent, $\xi = \phi_B - \phi_A$, relaxes rapidly via reaction kinetics. A systematic expansion exploits this separation of timescales, adiabatically eliminating $\xi$ to derive an effective hydrodynamic equation for $\psi$. The nonequilibrium character is encoded in a composition-dependent activity coefficient $\alpha(\psi, \xi)$ associated with the active reaction pathway.

The resulting macroscopic dynamics generalize the form of AMB+. The current for $\psi$ includes, beyond standard diffusive terms, non-equilibrium gradient-dependent contributions with coefficients ($\lambda_{\rm eff}$, $\zeta_{\rm eff}$, $\eta_{\rm eff}$) determined explicitly from the underlying microscopic parameters:

• $\lambda_{\rm eff}$: multiplicative term in $\xi = \phi_B - \phi_A$0
• $\xi = \phi_B - \phi_A$1: nonlinear, non-variational term proportional to $\xi = \phi_B - \phi_A$2
• $\xi = \phi_B - \phi_A$3: cubic gradient $\xi = \phi_B - \phi_A$4 term, recently identified as relevant in non-equilibrium coarse-graining
• $\xi = \phi_B - \phi_A$5: stabilizing higher-order gradient correction ($\xi = \phi_B - \phi_A$6), crucial when effective interfacial tension becomes negative

The crucial interfacial energy coefficient is renormalized by activity, $\xi = \phi_B - \phi_A$7, where $\xi = \phi_B - \phi_A$8 is explicitly computed from the reaction thermodynamics and the functional form of $\xi = \phi_B - \phi_A$9. When $\xi$0, the system can develop regular mesoscale structure via a finite-wavelength instability, a defining signature of active phase separation.

Numerical Implementation and Phase Phenomenology

A Flory-Huggins bulk free energy with specifically chosen interactions and activity parameters serves as a testbed. The dependence of the active coefficient $\xi$1 on solvent density generates a scenario where activity favors microphase separation at high dilution and destabilizes interfaces at higher densities.

Emergence of Microphases and Novel Dynamical States

Systematic simulations span both deterministic and stochastic regimes:

Figure 1: Snapshots at long times reveal microphase-separated states (dense droplets in dilute background), macrodroplet coarsening, and novel dynamic filamentous patterns depending on control parameters.

When $\xi$2, arrested microphases with periodic, monodisperse droplets emerge and persist (panel a, b, e, f in Figure 1). The droplet spacing and morphology are well predicted by the hydrodynamic theory. For higher global densities, the same conditions can result in dynamic, filament-like structures where interfaces undergo continuous elongation, fission, and migration (panel d, h in Figure 1). These patterns are reminiscent—but not identical—to active foam phases in AMB+, underlining the extended phenomenology due to higher-order nonlinearities in the hydrodynamic limit.

Asymmetric Ripening and Exotic Coarsening

Detailed ripening diagrams elucidate the conditions under which Ostwald ripening is suppressed or even reversed, producing two droplets of distinct size or stabilized microstructures (analogous to "reversed ripening"):

Figure 2: Ripening diagram showing the fate of size-asymmetric droplets for varying effective interfacial tension and chemical driving Δμ; microphase and reverse-Ostwald regimes are controlled by $\xi$3.

The precise location of the transition lines is matched by the analytically obtained criterion $\xi$4.

Nonequilibrium Entropy Production

Analysis of entropy production rates quantifies energy dissipation from both reaction and diffusive processes. The overall entropy production grows with increasing chemical driving and exhibits a pronounced kink at the onset of dynamic microphase formation (akin to a singularity in transport at a nonequilibrium phase transition):

Figure 3: Entropy production per unit area as a function of chemical driving Δμ, showing a clear kink at the transition to dynamic microphase/filament behavior, particularly visible in the diffusive contribution.

This result provides a direct, measurable link between non-equilibrium thermodynamics and emergent collective phase behavior in active emulsions.

Practical and Theoretical Implications

The developed framework establishes how the coarse-grained, field-theoretic description of active emulsions, parameterized directly by microscopic thermodynamic and kinetic inputs, reproduces the canonical nonequilibrium phases seen in AMB+ and extends them due to higher-order corrections.

• Direct connection to chemical/biological context: The identification of active coefficients with explicit reaction rates and energy transduction enables material-specific predictions for systems such as ATP-driven biomolecular condensates.
• Generality beyond traditional criticality: The existence of micro- and macro-phase separation, bubbly, and filamentous structures is shown to not require proximity to equilibrium critical points, nor a specific quartic free energy expansion.
• Design of dissipative soft materials: Control over reaction-diffusion length, chemical driving force, and interaction strength offers routes to engineer materials with robust, tunable self-organized architectures.
• Thermodynamic consistency and entropy accounting: The model enables calculation of the full entropy production along with its partition into diffusive and reaction channels, linking emergent dynamics to energy dissipation.

Outlook and Future Directions

Future developments include exhaustive mapping of the phase diagram for varied models of chemical activity, incorporation of additional thermodynamic fields (e.g., for multi-step biochemical networks or mechanochemical couplings), and investigation of spatiotemporal instabilities beyond stationary patterns. The approach lays the foundation for predictive modeling of non-equilibrium droplets in biological cells, synthetic active emulsions, and other complex fluids with coupled transport and reactions.

Conclusion

This paper systematically derives the hydrodynamic theory for chemically active emulsions, demonstrating that the generic features of active phase separation as captured by AMB+ emerge naturally from microscopic chemical kinetics and thermodynamics. The theory reproduces and extends the known landscape of exotic non-equilibrium phases, offers quantitative agreement with numerical simulations, and provides direct connections to measurable thermodynamic quantities such as entropy production. These advances open new avenues for understanding and controlling the self-organization of dissipative materials at mesoscopic and macroscopic scales.