Shortcomings of the bond orientational order parameters for the analysis of disordered particulate matter

P.

 

Steinhardt

,

D.

 

Nelson

, and

M.

 

Ronchetti

,  , ().

D.

 

Nelson

and

B.

 

Halperin

,  , ().

P.

 

ten Wolde

,

M.

 

Ruiz-Montero

, and

D.

 

Frenkel

,  , ().

R.

 

Ni

and

M.

 

Dijkstra

,  , ().

W.-S.

 

Xu

,

Z.-Y.

 

Sun

, and

L.-J.

 

An

,  , ().

W.

 

Lechner

and

C.

 

Dellago

,  , ().

L.-C.

 

Valdes

,

F.

 

Affouard

,

M.

 

Descamps

, and

J.

 

Habasaki

,  , ().

T.

 

Kawasaki

and

H.

 

Tanaka

,  , ().

H.

 

Wang

and

H.

 

Gould

,  , ().

Y.

 

Wang

,

S.

 

Teitel

, and

C.

 

Dellago

,  , ().

A.

 

Keys

and

S.

 

Glotzer

,  , ().

C.

 

Iacovella

,

A.

 

Keys

,

M.

 

Horsch

, and

S.

 

Glotzer

,  , ().

C.

 

Chakravarty

,

P. G.

 

Debenedetti

, and

F. H.

 

Stillinger

,  , ().

F.

 

Calvo

and

D. J.

 

Wales

,  , ().

J.

 

Hernández-Guzmán

and

E. R.

 

Weeks

,  , ().

K.

 

Binder

and

W.

 

Kob

, (, ).

A.

 

Ikeda

and

K.

 

Miyazaki

,  , ().

A. V.

 

Mokshin

and

J.-L.

 

Barrat

,  , ().

H.

 

Tanaka

,

T.

 

Kawasaki

,

H.

 

Shintani

, and

K.

 

Watanabe

,  , ().

K.

 

Lochmann

,

A.

 

Anikeenko

,

A.

 

Elsner

,

N.

 

Medvedev

, and

D.

 

Stoyan

,  , ().

T.

 

Schilling

,

H.

 

Schöpe

,

M.

 

Oettel

,

G.

 

Opletal

, and

I.

 

Snook

,  , ().

J. S.

 

van Duijneveldt

and

D.

 

Frenkel

,  , ().

A.

 

Wouterse

and

A. P.

 

Philipse

,  , ().

N.

 

Duff

and

D.

 

Lacks

,  , ().

S.

 

Abraham

and

B.

 

Bagchi

,  , ().

This expression assumes that neighborhood is a symmetric concept, such that

$a\in \protect \mathrm{NN}(b)$

$a∈ NN (b)$ implies that

$b\in \protect \mathrm{NN}(a)$

$b∈ NN (a)$⁠. This is correct for the definitions of neighborhood based on cutoff radii and on the Delaunay triangulation, but not for the definition based on a fixed number of neighbors.

C.

 

Kelchner

,

S.

 

Plimpton

, and

J.

 

Hamilton

,  , ().

S.

 

Edwards

and

D.

 

Grinev

,  , ().

M.

 

Bargiel

and

E. M.

 

Tory

,  , ().

P.

 

Armstrong

,

C.

 

Knieke

,

M.

 

Mackovic

,

G.

 

Frank

,

A.

 

Hartmaier

,

M.

 

Göken

, and

W.

 

Peukert

,  , ().

C.

 

Gray

and

K.

 

Gubbins

, (, , ).

E. P.

 

Wigner

, (, , ).

Although we will not use the global bond order parameter Q_l, we define it for the sake of completeness as |$Q_l = \sqrt {\frac{4\pi }{2l + 1} \sum_{m = - l}^l \big| \frac {1} {\mathcal{N}} \sum_{a = 1}^N \sum_{b \in {\rm NN} (a)}Y_{lm}(\theta_{ab}, \varphi_{ab})\big|^{2}} $|$Ql=4π2l+1∑m=−ll|1N∑a=1N∑b∈ NN (a)Ylm(θab,φab)|2$⁠. N is the number of spherical particles and |$\mathcal {N}=\sum_{a = 1}^N {n(a)}$|$N=∑a=1Nn(a)$ the number of all bonds. That is, the average over all bonds is taken inside the norm. For disordered systems the sum over the Y_lm vanishes as |$\mathcal{N}^{-1/2}$|$N−1/2$⁠, while it remains finite for common crystalline structures.^1,58

T.

 

Aste

,

M.

 

Saadatfar

, and

T.

 

Senden

,  , ().

B. A.

 

Klumov

,  , ().

M.

 

Yiannourakou

,

I. G.

 

Economou

, and

I. A.

 

Bitsanis

,  , ().

C. L.

 

Martin

,  , ().

S. C.

 

Kapfer

,

W.

 

Mickel

,

K.

 

Mecke

, and

G. E.

 

Schröder-Turk

,  , – ().

Normalized bond order functions are

$q_{lm}(a):=\big(\sum \nolimits _{i = 1}^{n(a)} {Y_{lm} }\big)/q_l(a)$

$qlm(a):=∑i=1n(a)Ylm/ql(a)$ for particle a and the dot-product is

$d_{ab}:=\sum \nolimits _{m=-l}^{l} q_{lm}(a)q_{lm}^*(b)$

$dab:=∑m=−llqlm(a)qlm*(b)$ of spheres a and b. A particle is defined as member of a solid-like cluster, if the dot-product with n₀ NN exceeds a certain threshold d₀.

A.

 

Mokshin

and

J.-L.

 

Barrat

,  , ().

A.

 

Panaitescu

and

A.

 

Kudrolli

,  , – ().

A. B.

 

de Oliveira

,

P. A.

 

Netz

,

T.

 

Colla

, and

M. C.

 

Barbosa

,  , ().

Z.

 

Yan

,

S. V.

 

Buldyrev

,

P.

 

Kumar

,

N.

 

Giovambattista

,

P.

 

Debenedetti

, and

H.

 

Stanley

,  , ().

M.

 

Wallace

and

B.

 

Joós

,  , ().

A.

 

Kansal

,

S.

 

Torquato

, and

F.

 

Stillinger

,  , ().

J. R.

 

Errington

and

P. G.

 

Debenedetti

,  , ().

G.

 

Odriozola

,  , ().

R.

 

Kurita

and

E.

 

Weeks

,  , ().

C. B.

 

Barber

,

D. P.

 

Dobkin

, and

H.

 

Huhdanpaa

,  , ().

The definition of NN via the Delaunay graph is equivalent to the definition via Voronoi neighbors: spheres share a Delaunay edge, whenever their respective Voronoi cells have a shared facet (regardless of the area of the Voronoi facet).

V.

 

Senthil Kumar

and

V.

 

Kumaran

,  , ().

S.

 

Matsumoto

,

T.

 

Nogawa

,

T.

 

Shimada

, and

N.

 

Ito

, “,” e-print arXiv:1005.4295 [cond-mat.stat-mech].

S. C.

 

Kapfer

,

W.

 

Mickel

,

F. M.

 

Schaller

,

M.

 

Spanner

,

C.

 

Goll

,

T.

 

Nogawa

,

N.

 

Ito

,

K.

 

Mecke

, and

G. E.

 

Schröder-Turk

,  , ().

Event driven MD simulations to explore the super-cooled regime use the Matsumoto algorithm from Ref. 52. In this algorithm, spheres are expanded until they touch the closest Voronoi facet or until they reach the final radius. This creates a transient polydisperse ensemble, which is relaxed by thermal motion, followed by an expansion step. This procedure is iterated until a monodisperse HS system at predefined packing fraction is obtained.

L. V.

 

Woodcock

,  , ().

P. R.

 

ten Wolde

,

M. J.

 

Ruiz-Montero

, and

D.

 

Frenkel

,  , ().

J. P.

 

Troadec

,

A.

 

Gervois

, and

L.

 

Oger

,  , ().

M. D.

 

Rintoul

and

S.

 

Torquato

,  , ().

G. E.

 

Schröder-Turk

,

W.

 

Mickel

,

S. C.

 

Kapfer

,

M. A.

 

Klatt

,

F. M.

 

Schaller

,

M. J. F.

 

Hoffmann

,

N.

 

Kleppmann

,

P.

 

Armstrong

,

A.

 

Inayat

,

D.

 

Hug

,

M.

 

Reichelsdorfer

,

W.

 

Peukert

,

W.

 

Schwieger

, and

K.

 

Mecke

,  , ().

R.

 

Schneider

and

W.

 

Weil

, , Teubner Skripten zur mathematischen Stochastik (, ).

G. E.

 

Schröder-Turk

,

W.

 

Mickel

,

M.

 

Schröter

,

G. W.

 

Delaney

,

M.

 

Saadatfar

,

T. J.

 

Senden

,

K.

 

Mecke

, and

T.

 

Aste

,  , ().

M.

 

Doi

and

T.

 

Ohta

,  , ().

G. E.

 

Schröder-Turk

,

V.

 

Trond

,

L. D.

 

Campo

,

S. C.

 

Kapfer

, and

W.

 

Mickel

,  , ().

M. E.

 

Evans

,

J.

 

Zirkelbach

,

G. E.

 

Schröder-Turk

,

A. M.

 

Kraynik

, and

K.

 

Mecke

,  , ().

S. C.

 

Kapfer

,

W.

 

Mickel

,

G. E.

 

Schröder-Turk

, and

K.

 

Mecke

, “,” (unpublished).

J.

 

Jerphagnon

,

D.

 

Chemla

, and

R.

 

Bonneville

,  , ().

C.

 

Lautensack

, (, , ).

H.

 

Müller

,  , ().

B. D.

 

Lubachevsky

and

F. H.

 

Stillinger

,  , ().

In the LS algorithm,^69,71 spheres are continuously expanded with event-driven MD until the pressure exceeds a jamming threshold (jLS). The unjammed LS simulations (uLS) used here are stopped at predefined packing fractions.

M.

 

Skoge

,

A.

 

Donev

,

F. H.

 

Stillinger

, and

S.

 

Torquato

,  , ().

The anisotropy index

$\beta _{\smash{1}}^{0,2}$

$β10,2$ is the ratio of eigenvalues of

$W_{\smash{1}}^{0,2}$

$W10,2$ (see Eq. (4)):

$\beta _{\smash{1}}^{0,2}=\xi _{{\min }}/\xi _{{\max }}$

$β10,2=ξmin/ξmax$⁠, where ξ_min ⩽ ξ_mid ⩽ ξ_max.

$\beta _{\smash{1}}^{0,2}=1$

$β10,2=1$ indicates isotropy, lower values of

$\beta _{\smash{1}}^{0,2}$

$β10,2$ indicate anisotropy.⁶¹ 

A.

 

Anikeenko

and

N.

 

Medvedev

,  , ().

W.

 

Mickel

,

G. E.

 

Schröder-Turk

, and

K.

 

Mecke

,  (), ().