Reciprocal theorem for ion-releasing colloidal particles

Technical Summary: Reciprocal Theorem for Ion-Releasing Colloidal Particles

Problem Statement
The migration of colloidal particles in electrolyte solutions, particularly those driven by surface chemical reactions (catalytic particles), presents a complex electro-hydrodynamic challenge. While the Lorentz reciprocal theorem is a standard tool for calculating propulsion speeds in low Reynolds number hydrodynamics, its application to catalytic particles has been assumed without rigorous justification. The core issue addressed in this paper is that catalytic particles releasing ions often generate a "secondary ionic cloud" distinct from the standard electrostatic diffuse layer (EDL). This secondary cloud, arising from the imbalance of ion diffusivities, imparts a net excess charge ($Q$) to the system. Existing theories, which typically assume electro-neutrality or neglect this secondary cloud, fail to account for the force exerted by external fields on this net charge. Consequently, the classical formulation of the reciprocal theorem may be insufficient for accurately predicting the mobility of both passive catalytic particles (migrating in external fields) and active microswimmers (self-propelling).

Methodology
The authors generalize the reciprocal theorem for particles suspended in electrolyte solutions subjected to either applied or spontaneously emerging electric fields. The methodology proceeds as follows:

1. Model Formulation: The study considers a spherical particle of radius $a$ in an aqueous salt solution. The fluid flow is governed by the Stokes equations, and the electrostatics by the Poisson equation. The ionic transport is described by the Nernst-Planck equations, neglecting convective fluxes (small Peclet number).
2. Reciprocal Theorem Derivation: An auxiliary problem is defined involving an inert, uncharged sphere translating with unit velocity. By applying the reciprocal theorem to the interaction between the actual catalytic system and this auxiliary system, the authors derive an integral relation for the particle velocity.
3. Inclusion of Far-Field Terms: A critical step involves evaluating the electrostatic force contribution at infinity ($S_\infty$). Unlike hydrodynamic stresses which decay rapidly, the electrostatic stress decays slowly due to the presence of the net system charge $Q$. The authors show that the electrostatic force includes a term proportional to the external field $E_\infty$ and the total charge $Q$.
4. Charge Calculation: The total charge $Q$ is calculated by solving the coupled Nernst-Planck and Poisson equations for the ion fluxes. By analyzing the asymptotic behavior of ion concentrations and potential at large distances, the authors derive an explicit expression for $Q$ in terms of the average surface ion flux $\langle J \rangle$, ion diffusivities ($D_+$, $D_-$), and bulk concentration.

Key Contributions
The primary contribution is the derivation of a generalized equation for the particle velocity ($V_p$) that explicitly includes the effect of the system's net charge:
$$V_p = \frac{1}{6\pi\eta a} \left[ \int_V \rho (\mathbf{D} - \mathbf{I}) \cdot \mathbf{E} \, dV + E_\infty Q \mathbf{e}_x \right]$$
where the second term represents the force on the secondary cloud.

The paper provides a specific analytical expression for the total charge $Q$:
$$Q = -\epsilon \beta \frac{a^2 k_B T}{e C_\infty D} \langle J \rangle$$
where $\beta = (D_+ - D_-)/(D_+ + D_-)$ is a parameter dependent on the difference in ion diffusivities, and $D$ is the harmonic mean diffusivity. This reveals that the sign of the excess charge is determined by the difference in ion diffusivities, while its magnitude is controlled by the average ion flux from the surface.

Results
The authors apply the generalized theorem to two specific scenarios:

1. Passive Catalytic Particles (Electrophoresis and Diffusiophoresis):

   • Electrophoresis: For particles migrating in an applied electric field, the results indicate that the contribution of the secondary cloud (the $Q$ term) is generally small. The mobility curves with and without the secondary cloud effect show only minor discrepancies, suggesting the classical theory remains largely valid for electrophoresis of catalytic particles.
   • Diffusiophoresis: For migration driven by salinity gradients, the effect is dramatic. The induced electric field is proportional to $\beta$. When $\beta \neq 0$, the secondary cloud significantly alters the mobility. Crucially, the paper demonstrates that the presence of the secondary cloud can not only change the magnitude of the mobility but also reverse its sign, a phenomenon not captured by theories assuming $Q=0$.

2. Active Microswimmers:

   • For self-propelling particles (Type I and Type II) in the absence of an external field ($E_\infty = 0$), the extra term vanishes, recovering the standard self-diffusiophoresis theory.
   • However, for Type II swimmers (which release both cations and anions non-uniformly) subjected to an external field, both the average flux $\langle J \rangle$ and the external field $E_\infty$ are non-zero. In this case, the generalized theorem predicts a significant modification to the swimmer's velocity and orientation dynamics.

Significance and Claims
The paper claims that the assumption of the classical reciprocal theorem's validity for catalytic particles is not obvious and requires generalization to account for the secondary ionic cloud. The significance of this work lies in:

• Theoretical Correction: It identifies a missing term in the force balance for ion-releasing systems, linking propulsion speed directly to the system's net charge and external fields.
• Predictive Power: It demonstrates that neglecting the secondary cloud can lead to qualitative errors, particularly in diffusiophoresis where the sign of mobility can be reversed.
• Broader Implications: The authors suggest that these results are relevant for understanding interactions between microswimmers, as the field induced by one swimmer acts as an external field for another. They posit that this framework could alter the understanding of collective behavior in active matter, though they note that detailed analysis of these interactions is reserved for future work.

The authors maintain a modest stance, noting that while the extra term vanishes or has little effect in specific electrophoretic scenarios, it may "dramatically change" the magnitude and sign of mobility in others, particularly for diffusiophoresis and active particles in external fields.