Ostwald ripening controlled by diffusion of a sparingly soluble component

Abstract: Additives of sparingly soluble components are known to slow down or completely inhibit Ostwald ripening in dispersed systems. In this paper, our earlier model of stabilization against Ostwald ripening is revisited and extended. In a quasi-steady-state mode, the process is shown to be controlled by the diffusion of the less soluble component, and the whole machinery of the classical Lifshits-Slezov-Wagner (LSW) theory can be leveraged almost without any change. The particle size distribution is predicted to follow the same distribution function pattern as in the classic LSW theory. The rate of ripening follows the classic cubic law. To extend our earlier result, an improved extrapolatory equation for the ripening rate is derived, that covers the whole formulation range, accounts for the difference in molar volumes of the components and for the solution non-ideality. The behavior described above is observed over the range of high concentrations of the poorly soluble component, with the cutoff determined by the lock-in number described in the previous paper of this series. When the concentration of the additive is low, the kinetics no longer follows the LSW pattern; instead, the particle size distribution becomes bimodal, with the fraction of 'fines' enriched by the poorly soluble component and the fraction of the large particles to ripen as if no additive were present. The lock-in parameter L1 can be used to characterize for the transition from one mode to another. In the end, some practical stabilization approaches for emulsions are discussed.

• The paper demonstrates that adding sparingly soluble components can arrest Ostwald ripening through a lock-in mechanism quantified by the L1 number.
• It employs advanced rate equations and numerical simulations to capture non-ideal solution effects and polydisperse initial conditions.
• Findings offer practical guidelines for stabilizing emulsions in pharmaceuticals, foods, and advanced material formulations.

Ostwald Ripening with Diffusional Control by Sparingly Soluble Components: Theoretical Extensions and Numerical Analysis

Introduction and Context

This study addresses Ostwald ripening in multicomponent emulsions, focusing specifically on systems where the dispersed phase contains not only a soluble main component but also a sparingly soluble additive. While classical Lifshitz–Slyozov–Wagner (LSW) theory successfully describes ripening kinetics and morphologies for single-component dispersed systems, it inadequately accounts for the pronounced stabilization empirically observed with small quantities of poorly soluble additives. The work systematically extends prior models, introduces improved rate equations valid across the entire composition range (considering different molar volumes and non-ideal solution effects), and employs numerical simulations for polydisperse initial conditions.

Classical LSW Framework and Its Limitations

The LSW theory quantitatively predicts the kinetics of Ostwald ripening in ideal, single-component systems via the cubic growth law:

$\frac{d\langle r\rangle^3}{dt} \propto D\,C_0\,a$

where $D$ is the solute diffusion coefficient, $C_0$ the bulk solubility, and $a$ an interfacial tension-dependent capillary length. The resulting particle size distribution is self-similar and time-invariant (LSW attractor), with a linear reduction in number density and a corresponding increase in mean particle size.

Experimentally, the addition of species with low solubility in the continuous phase strongly retards or even arrests ripening, which cannot be explained within the LSW theory. Earlier works (e.g., Higuchi–Misra, Webster–Cates) posited that the enrichment of fines with insoluble additives counteracts the Laplace pressure, suppressing the net driving force for mass transfer.

Two-Component Drop Model: Analytical Structure

The present analysis considers a dispersed phase composed of two mutually soluble species: a dominant, higher-solubility component (1), and a sparingly soluble additive (2). The model covers both ideal and non-ideal solutions, finite and vanishing solubilities, and incorporates differences in component molar volumes.

Central to the analysis is the “lock-in” number $L_1$ (generalizing prior single-particle stabilization criteria):

$L_1 = \frac{x_{02} r_0}{a_1 (1-x_{02})}$

with $x_{02}$ the initial molar fraction of the additive, $r_0$ the starting radius, and $a_1$ characterizing the Laplace pressure impact for component 1. $L_1$ governs the system’s qualitative evolution: large $D$0 leads to lock-in and stabilization, while small $D$1 results in failure to suppress ripening.

Kinetic Regimes and Morphological Outcomes

High Additive Concentration ($D$2): Lock-In and Kinetics

For sufficiently large $D$3, the Laplace pressure is balanced by the increase in additive concentration in shrinking particles (Raoult effect). The distribution does not coarsen, and mass transfer halts after an initial adjustment—this is the “lock-in” state. Numerical solutions reveal that for characteristic emulsions (e.g., $D$4, $D$5), the original distribution, whether LSW, log-normal, or gamma, remains essentially unchanged except for an increase in additive content among the smallest particles. The lock-in transition occurs with considerably minor compositional and size distribution evolution.

The generalization to finite solubility leads to a quasi-steady-state where the diffusion of the less soluble component (2) controls the ripening (if $D$6 and mass loss of 2 is slow). Here, an improved extrapolatory growth law is derived that smoothly interpolates between (i) diffusion-limited ripening controlled by the main component and (ii) the regime limited by the sparingly soluble additive:

$D$7

This result reduces to the LSW law in respective limits, incorporates non-idealities (activity coefficients), and applies to arbitrary compositions and molar volumes.

Low Additive Concentration ($D$8): Segregation and Bimodality

For low $D$9, the Raoult effect is insufficient to stabilize the system. The distribution evolves toward a bimodal shape: a persistent fines fraction heavily enriched in additive, and a fraction of larger particles coarsening according to classical LSW kinetics (cubic growth law). Notably, for realistic initial polydispersities (log-normal or gamma distributions), numerical simulations confirm this splitting and the persistence of “fines” (sub-40 nm) as a subpopulation.

In cases with slightly finite additive solubility, the fines are predicted to undergo slower, independent Ostwald ripening. Over extended timescales, mass exchange can occur from the fines back to the coarsening fraction, but this is markedly retarded compared to the main kinetic process.

Practical Stabilization Strategies and Parametric Insights

A key implication is that emulsion stabilization against Ostwald ripening can be tuned by increasing the molar fraction of a sparingly soluble additive or by reducing interfacial tension (thus decreasing $C_0$0 and facilitating lock-in at lower $C_0$1). Higher alkanes (C16–C18) or corresponding fats are effective additives within nanoscale droplet ranges. Very large polymeric hydrophobes, while less water-soluble, become less effective due to the decreasing $C_0$2 and thus a less favorable $C_0$3.

Lowering the oil–water interfacial tension by several orders of magnitude using optimized surfactants is a powerful, complementary lever, enabling stabilization with even smaller additive quantities for ultrasmall droplet sizes.

Implications and Future Directions

The results have direct utility for the formulation and shelf-life engineering of emulsions, particularly in pharmaceuticals, foods, and advanced material processing, where long-term stability is crucial. The unified kinetic law allows rational design across varying compositions, accounting for complex solution thermodynamics, diffusion dynamics, and realistic particle size distributions.

Theoretical implications include elucidation of previously ambiguous roles for “fines” subpopulations in droplet and polymerization emulsions, as well as in atmospheric aerosol bimodality, connecting disparate experimental findings under a coherent diffusive growth/stabilization mechanism. Although experimental validation of the model’s nuanced predictions (such as the precise law interpolating between kinetic regimes) requires further precision measurement, core features—lock-in, ripening arrest, bimodality—are well-supported by available data.

Future research may address the long-time dynamics for intermediate (finite) solubility cases in the $C_0$4 regime, the nonlinear coupling of nucleation and ripening in actively fragmenting systems, and the impact of solution nonidealities on transition thresholds and long-term emulsion evolution.

Conclusion

This paper extends the theory of Ostwald ripening in multicomponent systems by quantitatively analyzing the role of sparingly soluble additives. It systematically derives the conditions for kinetic stabilization (“lock-in”), introduces a comprehensive rate law valid across all composition ranges and solution nonideality, and verifies results with detailed simulations for realistic polydisperse systems. Emergence of fines and bimodal distributions is rationalized, and practical routes for stabilization via molecular and interfacial tuning are identified. The methodology and insights have significant implications for the formulation of stable dispersions and for understanding multiphase evolution in complex fluids.