Dissolution of a two-component drop onto macrophase due to surface tension effect

1. Problem Statement

The paper addresses the fundamental problem of the dissolution kinetics of a spherical, two-component liquid droplet into a surrounding medium (macrophase) driven by excess Laplace pressure (surface tension effects).

• Context: This work serves as a precursor to understanding Ostwald ripening in dispersed systems. It is well-known that adding sparingly soluble components can inhibit Ostwald ripening, but the mechanisms when the additive has finite solubility are not fully understood.
• Specific Challenge: The study focuses on a droplet composed of two miscible components (Component 1: highly soluble; Component 2: sparingly soluble). The goal is to model how the droplet dissolves, how its composition changes over time, and to derive a unified rate equation that covers the entire range of compositions (from pure Component 1 to pure Component 2) and accounts for non-ideal solution behavior and differences in molar volumes.

2. Methodology

The author employs a combination of thermodynamic analysis, diffusion kinetics, and numerical simulations.

• Theoretical Framework:
  • Mass Transport: The dissolution is modeled using steady-state diffusion equations (Fick's law) for spherical particles, defining fluxes ($J_1, J_2$) based on concentration gradients between the particle surface and the macrophase.
  • Chemical Potentials: The driving force is analyzed via excess chemical potentials ($\Delta \mu$), which include two competing effects:
    1. Laplace Effect: Driven by surface tension ($\sigma$) and curvature ($2\sigma/r$), promoting dissolution.
    2. Raoult Effect: Driven by changes in composition ($x_1, x_2$) and activity coefficients, which can counteract the Laplace effect.
  • Gibbs-Duhem Equation: Used to relate the chemical potential changes of the two components, allowing the model to handle non-ideal solutions without assuming ideality.
• Approximations:
  • The model assumes small relative variations in particle composition during the main dissolution phase (the "Lock-in" approximation).
  • It utilizes a "Steady State" approach similar to the Bodenstein method, assuming the chemical potential of the more soluble component is effectively zero ($\Delta \mu_1 \approx 0$) during the lock-in phase.
• Numerical Validation:
  • The author solves coupled differential equations numerically without the "Lock-in" approximation to validate the theoretical derivations.
  • Simulations vary key parameters: interfacial tension ($\sigma$), initial molar fractions ($x_{02}$), molar volumes ($V_{m1}, V_{m2}$), and solubilities.

3. Key Contributions and Findings

A. Identification of Three Dissolution Stages

The paper identifies three distinct kinetic regimes in the dissolution process:

1. Pre-lock-in: A rapid initial phase where the concentration of the sparingly soluble component ($x_2$) increases quickly. The Laplace pressure drives the dissolution of Component 1, causing $x_2$ to rise until the Raoult effect (due to high $x_2$) counterbalances the Laplace pressure.
2. Lock-in (Steady State): The dominant phase where the Laplace pressure effect on the more soluble component is nearly completely neutralized by the Raoult effect. The dissolution rate becomes controlled by the slow diffusion ("bleeding") of the sparingly soluble Component 2. During this stage, the particle composition remains nearly constant, and the volume decreases linearly with time (cubic law for radius).
3. Late Lock-in: Occurs near the end of the particle's life when the radius becomes very small. The Laplace pressure becomes so large that the Raoult effect can no longer fully counterbalance it. The composition changes rapidly, and the dissolution kinetics deviate from linearity (becoming quadratic in volume or deviating from the cubic law).

B. Definition of Dimensionless Numbers

Two critical dimensionless parameters are introduced to characterize the system:

• First Lock-in Number ($L_1$): Determines the validity of the lock-in approximation.
  • If $L_1 \gg 1$, the system enters a stable lock-in state, and the dissolution follows the classical cubic law ($r^3 \propto t$).
  • If $L_1 \approx 1$ or lower, the "late lock-in" effects become significant earlier, and the linear volume-time relationship breaks down.
• Second Lock-in Number ($L_2$): Relates to the ratio of solubilities and molar volumes, characterizing the "bleeding" rate of the sparingly soluble component.

C. Derivation of a Unified Dissolution Rate Equation

The paper proposes an improved empirical equation (Eq. 31) for the dissolution rate ($w$) that unifies the behavior across the entire composition range ($x_{02} = 0$ to $1$):
$$w = \frac{4\pi r D_2 C_2^*}{V_{m2}} \left[ \frac{\phi_1}{\phi_1^* C_1^* D_1 V_{m1}} + \frac{\phi_2}{\phi_2^* C_2^* D_2 V_{m2}} \right]^{-1}$$
(Note: The equation combines the rates of the two pure components weighted by their volume fractions and thermodynamic properties.)

• Significance: This equation extends the author's previous work (Ref [2]) by accounting for:
  • Differences in molar volumes ($V_{m1} \neq V_{m2}$).
  • Non-ideal solution behavior (activity coefficients).
  • The transition between pure-component dissolution and mixture dissolution.

4. Results and Validation

• Numerical Simulations:
  • High $L_1$ (e.g., $L_1 = 156$): The volume vs. time plot is perfectly linear, confirming the lock-in state. The theoretical slope matches the numerical simulation with high precision (correlation coefficient $\approx 1.0$).
  • Low $L_1$ (e.g., $L_1 = 1$): The linearity degrades, and the "late lock-in" concave deviation becomes visible. The dissolution rate deviates from the theoretical prediction based on Eq. 27.
  • Molar Volume Asymmetry: The new empirical equation (Eq. 31) accurately predicts dissolution rates even when molar volumes differ by a factor of 8. In contrast, the previous model (Eq. 36) failed significantly (up to 5x error) in these asymmetric cases.
• Bleeding Effect: When $L_2$ is low, the sparingly soluble component "bleeds" out too fast to establish a full lock-in. However, the kinetics still follow a cubic law, just at a different rate, rather than deviating from the law entirely.

5. Significance and Implications

• Theoretical Advancement: This paper provides a rigorous thermodynamic and kinetic framework for two-component drop dissolution, moving beyond idealized single-component models.
• Ostwald Ripening: The findings are directly applicable to stabilizing emulsions and dispersions. The "Lock-in" mechanism explains how sparingly soluble additives inhibit Ostwald ripening: by creating a steady-state barrier where the dissolution of the soluble component is halted by the accumulation of the insoluble component.
• Practical Application: The derived equation allows formulators to predict the stability and lifetime of complex multi-component droplets (e.g., in inkjet printing, pharmaceuticals, or cosmetics) by calculating the necessary concentration of sparingly soluble additives to achieve a specific $L_1$ value.
• Future Work: The author notes that while the $L_1 \gg 1$ case scales directly to Ostwald ripening theory (predicting standard particle size distributions), the $L_1 \approx 1$ case may lead to bimodal particle size distributions, a topic reserved for a follow-up paper.

In summary, Kabalnov successfully extends the theory of Ostwald ripening inhibition by quantifying the "lock-in" phenomenon, defining the conditions under which it occurs, and providing a robust, generalized equation for dissolution rates in non-ideal, two-component systems.