Morphometric Approach to Many-Body Correlations in Hard Spheres

Abstract

We model the thermodynamics of local structures within the hard sphere liquid at arbitrary volume fractions through the morphometric calculation of n-body correlations. We calculate absolute free energies of local geometric motifs in excellent quantitative agreement with molecular dynamics simulations across the liquid and supercooled liquid regimes. We find a bimodality in the density library of states where five-fold symmetric structures appear lower in free energy than four-fold symmetric structures, and from a single reaction path predict a relaxation barrier which scales linearly in the compressibility factor. The method provides a new route to assess changes in the free energy landscape at volume fractions dynamically inaccessible to conventional techniques.

Morphological thermodynamics for hard bodies from a controlled expansion

“…The free energy we have identified emerges rigorously as a contribution from the virial series, and its accuracy indicates that the success of morphological theories reported in previous investigations [4,6,[10][11][12][13] is enabled by this being a significant leading contribution. Moreover, the exact contribution provides a suitable starting point for including additional terms where improved accuracy is needed at e.g.…”

“…While not exact, this shows that the morphometric contributions account for ∼90% of the contributions to the equation of state. This fact suggests that the reported accuracy of morphological ther-modynamics for descriptions of the hard sphere liquid [4,6,[10][11][12][13] is possible because this exact contribution is a significant leading contribution. This is discussed in more detail in the context of FMT in [21], and is partially attributable to cancellations of terms omitted from the resummation.…”

Many-body correlations from integral geometry

“…In every formulation of scaled particle theory one considers a hard spherical solute of radius R. In most approaches, the cost ∆Ω is assumed to have an analytic expansion in powers of the radius; in classical approaches this was simply postulated, however we will be able provide proper justification below through geometric arguments. Recognising that terms scaling faster than R 3 must be zero for it to remain well-defined in the limit of large solutes leads to the third-order polynomial [16] ∆Ω(R) = p 4πR 3 3 + a 2 4πR 2 + a 1 4πR + a 0 4π, (6) where we identified the largest power with the work term pV from comparison with (5), and {a 0 , a 1 , a 2 } are thermodynamic coefficients describing the subleading corrections. We have chosen to introduce factors of 4π in front of the subleading terms to lead into the generalisation beyond spherical geometries.…”

“…where κ is the curvature tensor for the surface ∂K. For a spherical solute these reduce to the values given in (6), so this represents a proper generalisation of SPT for more general geometries. We give a brief justification of the ansatz (7) using integral geometric arguments in appendix A.…”

“…Proposed mechanisms for dynamical phenomena all loosely fall under the broad umbrella of manybody correlations; nucleation occurs via crystal seed formation [2], and to explain dynamic arrest approaching the glass transition thermodynamic theories invoke cooperatively rearranging regions [3] or elastic soft modes [4] while kinetic theories posit the existence of dynamical defects [5]. In a recent letter [6] we proposed a framework for treating many-body correlations, and developed an operational scheme for predicting the populations and dynamics of local structural motifs within a uniform liquid. Central to this is the use of the morphometric approach.…”