Spherical harmonic projections of the static structure factor of the dipolar hard sphere model: Theory vs simulations

Abstract

We investigate the static correlations of a dipolar fluid in terms of the irreducible coefficients of the spherical harmonic expansion of the static structure factor. To this end, we develop a theoretical framework based on a soft-core version of Wertheim’s solution of the mean spherical approximation (MSA), which renders the analytical determination of such coefficients possible. The accuracy of this approximation is tested by a comparison against the results obtained with the assistance of extensive molecular dynamics simulations at different regimes of concentration and temperature. Crucial aspects for the comparison of the results provided by the two methods are carefully discussed, concerning the different reference frames used in theory and simulations to describe rotations and orientations, and leading to important differences in the behavior of correlation functions with the same combination of spherical harmonic indices. We find a remarkable agreement between the two approaches in the fluid regime, thus providing a first stringent comparison of the irreducible coefficients of the spherical harmonic expansion of the dipolar fluid’s static structure factor, provided by the MSA theory and molecular dynamics simulations.

“Inner clocks” of glass-forming liquids

“…As discussed in Refs. 63-68, if all the orientations of particles, μa, and of the wave vector, μk = k/|k|, are described in the so-called k-frame, 65,66 one is left with correlation functions of the form F ll ′ m (k, τ; t) ≡ F lm;l ′ m ′ (k, τ; t)δ mm ′ , F S ll ′ m (k, τ; t) ≡ F S lm;l ′ m ′ (k, τ; t)δ mm ′ , and S ll ′ m (k; t) ≡ S lm;l ′ m ′ (k; t)δ mm ′ . Furthermore, and on the basis of previous work for the specific case of the DHS fluid, 65,66 one might restrict to consider only diagonal (l = l ′ ) components F lm (k, τ; t) ≡ F ll ′ m (k, τ; t)δ ll ′ , F S lm (k, τ; t) ≡ F S ll ′ m (k, τ; t)δ ll ′ , and S lm (k; t) ≡ S ll ′ m (k; t)δ ll ′ .…”

“…63-68, if all the orientations of particles, μa, and of the wave vector, μk = k/|k|, are described in the so-called k-frame, 65,66 one is left with correlation functions of the form F ll ′ m (k, τ; t) ≡ F lm;l ′ m ′ (k, τ; t)δ mm ′ , F S ll ′ m (k, τ; t) ≡ F S lm;l ′ m ′ (k, τ; t)δ mm ′ , and S ll ′ m (k; t) ≡ S lm;l ′ m ′ (k; t)δ mm ′ . Furthermore, and on the basis of previous work for the specific case of the DHS fluid, 65,66 one might restrict to consider only diagonal (l = l ′ ) components F lm (k, τ; t) ≡ F ll ′ m (k, τ; t)δ ll ′ , F S lm (k, τ; t) ≡ F S ll ′ m (k, τ; t)δ ll ′ , and S lm (k; t) ≡ S ll ′ m (k; t)δ ll ′ . Within these considerations, the NE-SCGLE theory is summarized by a set of coupled evolution equations for the waiting-time dependent functions involved, whose solution for t > 0 describes the irreversible evolution of the structural and dynamical properties of an instantaneously quenched DHS liquid.…”

“…( 27) and has been systematically employed 62 to understand the dynamically arrested states already detected in extensive molecular dynamics (MD) simulations of this model. 65,66 We summarize the relevant ingredients of the non-spherical version of the NE-SCGLE theory in the Appendix.…”

“…Similar to the spherical case discussed previously, the most central equation of the extended NE-SCGLE describes the timeevolution of the non-equilibrium structure factor, which for liquids involving non-spherical interactions can be defined as S(k, μ; k ′ , μ ′ ; t) ≡ ⟨δn(k, μ; t) δn * (k ′ , μ ′ ; t)⟩. Here, δn(k, μ; t) is the Fourier transform of the fluctuation in the local number density n(r, μ; t) ≡ N −1/2 ∑ N a=1 δ(r − ra(t))δ(μ − μa(t)) of particles at the position r and with orientation μ at time t. 67 In practice, however, one focuses on the coefficients of the spherical harmonic expansion of n(k, μ; t), 63,[65][66][67][68] commonly referred to as tensorial density modes and denoted as nlm (k, t) [see Eq. (A1)].…”

“…(A1)]. These coefficients, in turn, allow us to define the corresponding projections of S(k, μ; k ′ , μ ′ ; t), namely, 65,66…”