Electrostatic–Elastic Softening and Ultraviolet Instability Driven by Non-DLVO Interactions in Charged Colloidal Crystals

Colloidal crystals permeated by mobile ions exhibit a coupling between electrostatic and elastic degrees of freedom that renormalizes the effective screening length and induces wave-vector-dependent elastic softening. Building on our recently proposed continuum model, we perform a rigorous Gaussian fluctuation analysis to elucidate the stability limits of the homogeneous phase. By integrating out the electrostatic fluctuations, we derive the effective elastic modulus Γ(q) as a function of wave vector q. We show that the modulus in the long-wavelength limit (q→0) remains identically equal to a bare modulus protected by perfect ionic screening. In contrast, the modulus in the short-wavelength limit (q→∞) softens as the electrostatic-elastic coupling strength ξ increases, vanishing at a critical value ξ=1. For ξ>1, the fluctuation spectrum exhibits a negative eigenvalue for all wave vectors q larger than a critical (effective screening) wave vector qc, signaling an ultraviolet instability of the uniform phase. In a real colloidal crystal, this divergence is regulated by the discrete lattice cutoff qmax∼π/a, confining the physical instability to a finite band qc