Asymptotic decay of correlations in liquids and their mixtures

Abstract

We consider the asymptotic decay of structural correlations in pure fluids, fluid mixtures, and fluids subject to various types of inhomogeneity. For short ranged potentials, both the form and the amplitude of the longest range decay are determined by leading order poles in the complex Fourier transform of the bulk structure factor. Generically, for such potentials, asymptotic decay falls into two classes: (i) controlled by a single simple pole on the imaginary axis (monotonic exponential decay) and (ii) controlled by a conjugate pair of simple poles (exponentially damped oscillatory decay). General expressions are given for the decay length, the amplitude, and [in class (ii)] the wavelength and phase involved. In the case of fluid mixtures, we find that there is only one decay length and (if applicable) one oscillatory wavelength required to specify the asymptotic decay of all the component density profiles and all the partial radial distribution functions gij(r). Moreover, simple amplitude relations link the amplitudes associated with the decay of correlation of individual components. We give explicit results for the case of binary systems, expanding on and partially correcting recent work by Martynov. In addition, numerical results for g(r) for the pure fluid square-well model and for gij(r) for binary hard sphere mixtures are presented in order to illustrate the fact that the asymptotic forms remain remarkably accurate at intermediate range. This is seen to arise because the higher order poles are typically well-separated from the low order ones. We also discuss why the asymptotics of solvation forces for confined fluids and of density profiles of inhomogeneous fluids (embracing wetting phenomena) fall within the same theoretical framework. Finally, we comment on possible modifications to the theory arising from the presence of power-law attractive potentials (dispersion forces).

Density functional theory for a model colloid polymer mixture: bulk fluid phases

“…In the neighbourhood of the critical point g i j (r ) will decay monotonically, as befits OZ behaviour, whereas for small values of η p,r the mixture is HS-like and the g i j (r ) should exhibit damped oscillatory decay as r → ∞. Thus upon varying the thermodynamic parameters η p,r and η c the ultimate decay of g i j (r ) should change from being oscillatory to purely monotonic [42][43][44][45]. The line in the phase diagram separating the two types of decay is termed the FW line [42] after the authors who introduced the concept.…”

“…Oscillatory behaviour, h i j (r → ∞) ∝ cos(a 1 r ) exp(−a 0 r )/r , stems from poles with non-zero real part a 1 , whereas monotonic behaviour, h(r → ∞) ∝ exp(−a 0 r )/r , stems from poles residing on the imaginary axis, a 1 = 0. The ultimate decay is governed by the pole with the smallest imaginary part a 0 [44]. We obtain the location of the singularities by finding the roots of 1/|S i j (k)| = 0 numerically, taking appropriate starting values.…”

Entropic Wetting and the Free Isotropic−Nematic Interface of Hard Colloidal Platelets

“…We investigate in detail the results from the respective approaches for the asymptotic decay of density profiles at large distances from a wall. [45][46][47][48][49] A short account of this work has been published as part of ref 50.…”

Screening properties of four mesoscale smoothed charge models, with application to dissipative particle dynamics

“…The asymptotic behaviour of h ±± (r) is determined by the positions of the poles ofh ±± (k), regarded as analytic functions in the complex k-plane [9,19,20]. The functions share a common set of poles.…”