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 a recently proposed continuum model [\textit{Commun. Theor. Phys.} \textbf{77}, 055602 (2025)], 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 $\Gamma(q)$ as a function of wave vector $q$. We show that the long-wavelength modulus $\Gamma(0)$ remains identically equal to the bare modulus $\beta K$, protected by perfect ionic screening. In contrast, the short-wavelength modulus $\Gamma(q\to\infty) = \beta K(1-\xi)$ softens as the electrostatic-elastic coupling $\xi \equiv 2\beta n_0 v_0^2 K$ increases, vanishing at a critical value $\xi=1$. For $\xi>1$, the fluctuation spectrum exhibits a negative eigenvalue for all wave vectors $q > q_c = \kappa_0/\sqrt{\xi-1}$, signaling an ultraviolet instability of the uniform phase. In a real colloidal crystal, this divergence is regulated by the discrete lattice cutoff $q_{\max}\sim\pi/a$, confining the physical instability to a finite band $q_c < q < q_{\max}$. The macroscopic limit $q\to 0$ remains unconditionally stable for all $\xi$. The transition at $\xi=1$ thus marks the onset of short-wavelength mechanical failure, while macroscopic elastic stiffness remains intact. Our analysis clarifies the proper physical interpretation of the minimal coupling model and provides a consistent picture of how non-DLVO interactions can drive local structural collapse in charged colloidal crystals.