Gelation as a Dynamical Instability of the Smoluchowski Flow

Area of Science:

• Physical Chemistry
• Chemical Engineering
• Statistical Mechanics

Background:

• Gelation is traditionally viewed as a singularity in coagulation equations.
• Existing models often focus on mass loss or moment divergence.
• A dynamical perspective on gelation is needed.

Purpose of the Study:

• To re-characterize gelation not as a singularity but as a loss of dynamical stability.
• To identify a spectral signature preceding gel formation.
• To unify aggregation dynamics within a broader phase transition framework.

Main Methods:

• Analysis of the Smoluchowski coagulation equation with homogeneous kernels (K(i,j) = (ij)α).
• Quantification of dynamical stability using the time-dependent spectrum of the Jacobian.
• Comparison of gelling and nongelling kernels.

Main Results:

• Gelation is consistently preceded by the emergence of positive real eigenvalues, indicating a loss of local dynamical stability.
• Gelling kernels exhibit persistent spectral destabilization linked to macroscopic gel formation.
• Nongelling kernels show only transient finite-size effects.

Conclusions:

• Gelation can be understood as a loss of dynamical stability in aggregation processes.
• A spectral signature reliably predicts and characterizes gelation.
• This work unifies gelation kinetics with nucleation phenomena in a dynamical framework.