Self-consistent generalized Langevin-equation theory for liquids of nonspherically interacting particles

Abstract

A self-consistent generalized Langevin-equation theory is proposed to describe the self- and collective dynamics of a liquid of linear Brownian particles. The equations of motion for the spherical harmonics projections of the collective and self-intermediate-scattering functions, F_{lm,lm}(k,t) and F_{lm,lm}^{S}(k,t), are derived as a contraction of the description involving the stochastic equations of the corresponding tensorial one-particle density n_{lm}(k,t) and the translational (α=T) and rotational (α=R) current densities j_{lm}^{α}(k,t). Similar to the spherical case, these dynamic equations require as an external input the equilibrium structural properties of the system contained in the projections of the static structure factor, denoted by S_{lm,lm}(k). Complementing these exact equations with simple (Vineyard-like) approximate relations for the collective and the self-memory functions we propose a closed self-consistent set of equations for the dynamic properties involved. In the long-time asymptotic limit, these equations become the so-called bifurcation equations, whose solutions (the nonergodicity parameters) can be written, extending the spherical case, in terms of one translational and one orientational scalar dynamic order parameter, γ_{T} and γ_{R}, which characterize the possible dynamical arrest transitions of the system. As a concrete illustrative application of this theory we determine the dynamic arrest diagram of the dipolar hard-sphere fluid. In qualitative agreement with mode coupling theory, the present self-consistent equations also predict three different regions in the state space spanned by the macroscopic control parameters η (volume fraction) and T* (scaled temperature): a region of fully ergodic states, a region of mixed states, in which the translational degrees of freedom become arrested while the orientational degrees of freedom remain ergodic, and a region of fully nonergodic states.

Equilibration and Aging of Liquids of Non-Spherically Interacting Particles

“…For the same reason, Eqs. (47) and (48) imply that the normalized intermediate scattering functions F 00 (k, τ ; t)/S 00 (k; t) and F S 00 (k, τ ; t) will be unity for all positive values of the correlation time τ and waiting time t. 1 (τ ; φ, T i ) and the (blue) dot-dashed line is the asymptotic limit C 1 (τ ; t → ∞) = C (eq) 1 (τ ; φ, T f ). The inset plots the α-relaxation time, defined as C 1 (τ α ; t) = 1/e, as a function of waiting time t.…”

“…In this section we briefly describe the equilibrium SCGLE theory of the dynamics of liquids formed by non-spherical particles developed by Elizondo-Aguilera et al [48]. We first describe the main properties involved in this description and then summarize their time-evolution equations, which constitute the essence of the SCGLE theory.…”

“…In Ref. [48] the generalized Langevin equation (GLE) formalism and the concept of contraction of the description were employed to derive exact memory function equations for F (eq) lm;l ′ m ′ (k, τ ) and F S(eq) lm;l ′ m ′ (k, τ ). These dynamic equations only involve the corresponding projections S (eq) lm;l ′ m ′ (k) of the equilibrium static structure factor.…”

“…[48], the referred exact memory function equations for F (eq) lm;l ′ m ′ (k, τ ) and F S(eq) lm;l ′ m ′ (k, τ ) require the independent determination of the corresponding self and collective memory functions. In a manner similar to the spherical case, simple Vineyard-like approximate closure relations for these memory functions convert the originally exact equations into a closed self-consistent system of approximate equations for the dynamic properties referred to above [48]. These equations thus constitute the extension of the equilibrium SCGLE theory of the dynamic properties of liquids whose particles interact through nonspherical pair potentials.…”

“…In the simplest version (we refer the reader to Ref. [48] for details) these equations involve only the diagonal elements F lm (k, τ ) ≡ F lm;lm (k, τ ) and F S lm (k, τ ) ≡ F S lm;lm (k, τ ), and are written, in terms of the corresponding Laplace transforms F lm (k, z) and F S lm (k, z), as…”