Non-hyperuniform metastable states around a disordered hyperuniform state of densely packed spheres: stochastic density functional theory at strong coupling

Abstract

Disordered and hyperuniform structures of densely packed spheres near and at jamming are characterized by vanishing of long-wavelength density fluctuations, or equivalently by long-range power-law decay of the direct correlation...

Electric-field-induced oscillations in ionic fluids: a unified formulation of modified Poisson–Nernst–Planck models and its relevance to correlation function analysis

“…It has been shown that the constrained free energy as a functional of given density fields n ( r , t ) = ( n 1 ( r , t ), n 2 ( r , t )) T can be expressed by considering fluctuating potential fields ϕ ( r , t ) = ( ϕ 1 ( r , t ), ϕ 2 ( r , t )) T , which are conjugate to n ( r , t ), in addition to an adjusted potential field ϕ dft l ( r , t ) similar to that of the equilibrium DFT. 37,38 Extending the previous result 24,28 to the expression for two-component systems (see Appendix B for details), we havewith the following constraint imposed by the canonical ensemble:where the total number of either anions or cations is equally N . The free-energy functional F [ n , ϕ ] in the exponent of eqn (A3) is defined using the grand potential of the primitive model with an imaginary external field i ϕ ( r ) applied, and can be divided into two parts (see Appendix B for details): F [ n , ϕ ] = F [ n ,0] + Δ F [ n , ϕ ].The free-energy functional F [ n ,0] in the absence of fluctuating potential reduces to the intrinsic Helmholtz free energy, a key thermodynamic quantity in the equilibrium DFT.…”

“…Details on the constrained free energy of a given density distribution n ( r , t )We consider the overdamped dynamics of ions with the total number N of charged spheres being fixed. Hence, the constrained free-energy functional of a given density distribution n l ( r , t ) ( l = 1,2) is defined for the canonical ensemble using the contour integral over a complex variable w = e μ : 28 In eqn (B1), the Dirac delta functional represents the constraint on the original density distribution . It has been shown for one-component fluid that the constrained free energy functional is expressed by the functional integral over a fluctuating potential field.…”

“…The Dean–Kawasaki equation can be linearized for fluctuating density field around a reference density. 13–15,27,28 The linearized Dean–Kawasaki equation has proved relevant to describe various dynamics. It is an outstanding feature of the linearized SDFT to justify the inclusion of stochastic processes into the PNP model.…”

“…Recently, the stochastic mPNP models based on the linearized SDFT have provided the following results: 13–16,27,28 First, the linear Dean–Kawasaki equation has formulated ion concentration-dependent electrical conductivity. The obtained expression for conductivity reproduces the Debye–Hückel-Onsager theory and also explains the experimental results on concentrated electrolytes where the Debye–Hückel-Onsager theory breaks down.…”

“…It should also be remembered that the stochastic mPNP models have been concerned with linear response dynamics from a uniform density distribution and that there have been few systematic studies on an inhomogeneous density distribution in a steady state. 28 Further modifications of the stochastic mPNP models need to be made for incorporating improvements in the deterministic mPNP models.…”