Structure of saturated RSA ellipse packings Pedro Abritta and Robert S. Hoy Department of Physics University of South Florida Tampa FL 33620 USA

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Structure of saturated RSA ellipse packings

Pedro Abritta and Robert S. Hoy

Department of Physics, University of South Florida, Tampa, FL 33620 USA∗

(Dated: October 25, 2022)

Motivated by the recent observation of liquid glass in suspensions of ellipsoidal colloids, we ex-

amine the structure of (asymptotically) saturated RSA ellipse packings. We determine the packing

fractions φs(α) to high precision, ﬁnding an empirical analytic formula that predicts φs(α) to

within less than 0.1% for all α≤10. Then we explore how these packings’ positional-orientational

order varies with α. We ﬁnd a transition from tip/side- to side/side-contact-dominated structure at

α=αTS '2.4. At this aspect ratio, the peak value gmax of packings’ positional-orientational pair

correlation functions is minimal, and systems can be considered maximally locally disordered. For

smaller (larger) α,gmax increases exponentially with deceasing (increasing) α. Local nematic order

and structures comparable to the precursor domains observed in experiments gradually emerge as α

increases beyond 3. For α&5, single-layer lamellae become more prominent, and long-wavelength

density ﬂuctuations increase with αas packings gradually approach the rod-like limit.

I. INTRODUCTION

Recent advances in colloidal synthesis and microscopy

techniques have dramatically improved our ability to

characterize how particle ordering and relaxation in ther-

mal liquids varies with particle shape. For example, high-

quality monodisperse colloidal rods and cylinders can

now be produced [1], and their liquid-state dynamics (in

suspensions) can be observed using confocal microscopy

and allied techniques [2, 3]. Of particular interest is the

recent experimental observation of “liquid glass”. Roller

et al. found [4] that suspensions of ellipsoidal colloids

with aspect ratio αR= 3.5 exhibited two distinct glass

transitions at packing fractions φrot

gand φtrans

g. In the

liquid glass state (φrot

g< φ < φtrans

g), particles rotations’

are arrested, but they remain free to translate within

locally-nematic precursor domains. This phenomenon

had been predicted by simulations [5], but had previously

only been experimentally observed in quasi-2D systems

[6–8]. Experiments like Ref. [4] oﬀer both a new avenue

for understanding the physics of anisotropic molecular

glassformers and an obvious motivation for theoretical

studies of related models.

For hard ellipses and ellipsoids of revolution with as-

pect ratio α, complex liquid-state dynamics are expected

for packing fractions in the range φo(α)< φ < φg(α),

where φo(α) is the “onset” density [9, 10] and φg(α) is

either the rotational or translational glass transition den-

sity. Evaluating φo(α) and φg(α) using simulations is

very computationally expensive [11–14]. A more eas-

ily obtained yet physically relevant packing fraction that

lies between φo(α) and φg(α) is the random sequential

adsorption (RSA) density φs(α), the maximum density

at which impenetrable ellipses of aspect ratio αcan be

packed under a protocol that sequentially inserts them

with random positions and orientations. The diﬀerences

φg(α)−φs(α) and φJ(α)−φs(α) are of particular inter-

est because they indicate how much packing eﬃciency

∗rshoy@usf.edu

can be gained (before glass formation and jamming, re-

spectively) by allowing particles to move freely while re-

maining positionally and orientationally disordered. For

example, in the α→ ∞ limit one expects φJ(α) = φs(α)

because particle rotations are completely blocked [15, 16].

While jamming of ellipses and ellipsoids is now fairly

well understood [17–20], RSA of these systems remains

relatively poorly characterized. φs(α) values for ellipses

have been reported only for 1 ≤α≤5 [21–24], and de-

tailed characterization of RSA ellipse packings’ structure

has only been performed for the aspect ratio α'1.85

that maximizes φs[23]. Thus there is a need to substan-

tially expand our knowledge of these packings.

In this paper, we characterize saturated RSA ellipse

packings over a wider range of aspect ratios and in much

greater detail than has been previously attempted. First

we determine their packing fractions φs(α) to within

∼0.1% for 1 ≤α≤10. Then we characterize their

positional-orientational order using several metrics. We

ﬁnd a previously-unreported structural transition at α=

αTS '2.4. For 1 < α < αTS (α > αTS), packings have

an excess of tip-to-side (side-to-side) contacts. The peak

prevalence of the favored type of contact, an eﬀective

order parameter for these systems, increases exponen-

tially with |α−αTS|. We also show that (i) local ne-

matic order and structures comparable to the precursor

domains observed in experiments [4, 6] gradually emerge

as αincreases beyond 3, and (ii) the increasing size of

single-layer lamellae that are randomly oriented within

these packings makes their long-wavelength density ﬂuc-

tuations increase rapidly with αfor α&5.

II. GENERATING SATURATED PACKINGS

Saturated RSA packings of anistropic 2D particles are

generated by placing them with random positions and

orientations, typically within square domains of size L×

L, until no more particles can be inserted. In practice,

RSA packing generation’s inherently slow kinetics [25]

make achieving complete saturation for Lthat are large

arXiv:2210.02388v2 [cond-mat.soft] 24 Oct 2022

enough to minimize ﬁnite-size eﬀects extremely diﬃcult.

Therefore we employ an eﬃcient protocol that generates

packings that are demonstrably within <1% (and for

most α, much less than 1%) of their saturation densities,

for system sizes that are in the L→ ∞ limit. Speciﬁcally,

we use L= 1000σ, where σis the length of the ellipses’

minor axes. This is the same Lemployed in Refs. [23, 24]

and is suﬃciently large that ﬁnite-size errors on φsshould

be less than 0.1% [23]. Below, we set σ= 1.

We divide our periodic domains into Λ = ﬂoor(L/α)2

linked cells of size (L/Λ) ×(L/Λ). Here ﬂoor[x] rounds

xdownward to the nearest integer, e.g. ﬂoor[4.6] = 4.

Thus, when insertion of an ellipse iat position ~riis at-

tempted, only ellipses in the cell containing ~riand its

8 neighboring cells need be checked for overlap. Checks

for overlap with the set {j}of all ellipses that have al-

ready been inserted in these cells (at positions {~rj}) in-

clude three steps: (1) Since we assume ellipses’ minor

axes have unit length while their major axes have length

α, any center-to-center distances rij =|~rj−~ri|<1 im-

ply overlap, and the insertion attempt is rejected. (2) If

rij > α, the ellipses do not overlap and the code continues

to the next j. (3) If 1 ≤rij ≤α, overlap is possible. We

determine whether the ellipses overlap using Zheng and

Palﬀy-Muhoray’s exact expression [26] for their distance

of closest approach dcap. For monodisperse ellipses with

unit-length minor axes, their expression reduces to

dcap =d0

q1−(1 −α−2)( ˆ

ki·ˆrij )2

,(1)

where d0is obtained using a complicated formula involv-

ing rij and the ellipses’ orientation vectors ˆ

kiand ˆ

kj.

Note that ˆ

kjdoes not appear in Eq. 1 because it has

been absorbed into d0; cf. Eqs. 33, 36 of Ref. [26]. If

dcap > rij , the ellipses overlap and the insertion attempt

is rejected. Otherwise the ellipse is inserted and the next

insertion attempt begins.

We examined 81 diﬀerent particle aspect ratios over

the range 1 ≤α≤10. To allow extrapolation of the

runs’ progress to the inﬁnite-time limit, we tracked the

packing fractions φ(α, t) after tinsertion attempts per

unit area had been completed. Each run continued until

2.5×105trials per unit area had been attempted; this was

suﬃcient to reach well into the asymptotic t−1/3regime

[25] for all α. Then we determined the RSA densities

φs(α) by ﬁtting the results to

φ(α, t) = φs(α) 1−t

τ(α)−1/3!,(2)

where τ(α) is a “time” characterizing the α-dependent

RSA kinetics.

Figure 1 shows φ(α, t) for selected α. There is some

crossing of the curves at intermediate tfor α < 2 because

φs(α) is non-monotonic, but as expected for anisotropic

particles [27], τincreases with α. All α≤10 had τ < 0.13

and hence had φs(α)/φ(α, 2.5×105)−1< .0082. In

α=1.30 α=10

1 10 100 1000 104105

0.30

0.60

FIG. 1. RSA kinetics for ellipses. Results are shown for α=

1.3, 1.85, 2.4, 3.5, 5, 6.5, 8, and 10.

other words, at the end of our packing-generation runs,

all α≤10 have φthat are within less than 1% of φs(α).

III. STRUCTURE OF SATURATED PACKINGS

Figure 2 shows the extrapolated φs(α). Results for all

α≤5 agree with previous studies. In particular, they are

consistent with the well-known disk RSA packing fraction

φs(1) = φdisks =.54707 [28, 29] and the known density

maximum at α'1.85, i.e. φs(1.85) '.584 [21–24]. For

large aspect ratios, our results show a surprisingly slow

decrease of φswith increasing α. Speciﬁcally, since the

area swept out by ellipses as they rotate about their cen-

ters is Asw =πα2/4 whereas the area of the ellipses them-

selves is πα/4, Onsager’s classical arguments [30] predict

φs∼A/Asw = 1/α. Our data show that this asymptotic

regime is not reached until α10.

1 2 4 6 8 10

0.45

0.5

0.55

0.6

ϕs

FIG. 2. Packing fractions of saturated RSA ellipse pack-

ings.. Red circles show the φs(α) estimated from Eq. 2, while

the blue curve shows Eq. 3. The dashed gray line shows

φ= 0.775α−1/5and is included only to illustrate the lack

of convergence to α−1scaling.

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StructureofsaturatedRSAellipsepackingsPedroAbrittaandRobertS.HoyDepartmentofPhysics,UniversityofSouthFlorida,Tampa,FL33620USA(Dated:October25,2022)Motivatedbytherecentobservationofliquidglassinsuspensionsofellipsoidalcolloids,weex-aminethestructureof(asymptotically)saturatedRSAellipsepackings.Wedet...

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Structure of saturated RSA ellipse packings Pedro Abritta and Robert S. Hoy Department of Physics University of South Florida Tampa FL 33620 USA

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