硬粒子模型解析：凝聚相物质的密集堆积研究进展

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"《观点：通过堆积模型理解凝聚相物质的基本特性》这篇文章探讨了数千年来的研究热点——堆积问题及其在多个学科领域的广泛影响。文章着重于硬粒子堆积模型的研究，这些模型为我们提供了对凝聚相物质结构和宏观性质的基本认识，包括低温态（如分子和胶体液体、晶体以及玻璃态）、多相异质介质、颗粒系统以及生物体系。密集堆积在纯数学中，特别是离散几何和数论中具有极高的兴趣，它们与最大密度、最随机堆积以及最低密度稳定的结构密切相关。作者S. Torquato回顾了关于在不同欧几里得空间维度中，平衡、亚稳和非平衡硬粒子堆积的理论和计算文献。其中，对于被“阻塞”或“ jammed”堆积的分析，采用了“几何结构”方法，这是一种强大的工具，它通过jamming类别和“秩序”映射来定量刻画单个堆积，涵盖了极限阻塞状态，如最密堆积、最大随机阻塞状态以及最低密度稳定结构。文章还涵盖了同质球体、大小分布不均的球体以及非球形粒子的堆积情况。作者在总结时指出了未来研究的挑战和未解问题，鼓励进一步探索这些领域，以深化我们对凝聚相物质的理解。本文发表于美国物理学会，链接可供读者查阅。此外，文章还提到了与《化学物理学杂志》相关的一些其他观点文章，如电子摩擦的理解、多参考耦合簇理论动态电子关联、环聚合物瞬态理论、机器学习在原子尺度模拟中的应用，以及计算机化学软件的发展等，这些都反映了该领域内交叉学科的最新进展。通过这些视角，读者可以更全面地了解化学物理学中的前沿思考和研究动态。"

020901-5 S. Torquato J. Chem. Phys. 149, 020901 (2018)

which can be regarded to be glasses. A rapid compression leads

to a lower random jammed density than that for a slow com-

pression. The most rapid compression ending in mechanically

stable packing is presumably the maximally random jammed

(MRJ) state

with φ ≈ 0.64. Accurate approximate formu-

las for the pressure along such metastable extensions up to

the jamming points have been obtained.

20,83

This ideal amor-

phous state is described in greater detail in Sec. III E. Note

that rapid compression of a hard-sphere system is analogous

to supercooling of a molecular liquid.

Pair statistics are exactly known only in the case of 1D

“hard-rod” systems.

For d ≥ 2, approximate formulas for

(r) are known along liquid branches.

Approximate clo-

sures of the Ornstein-Zernike integral equation linking the

direct correlation function c(r) to the total correlation func-

tion h(r),

such as the Percus-Yevick (PY) and hypernetted

chain schemes,

2,7,12,86

provide reasonably accurate estimates

of g

(r) for hard-sphere liquids. Because g

(r) decays to unity

exponentially fast for liquid states, we can conclude that it

must have a corresponding structure factor S(k) that is an even

function and analytic at k = 0; see Eq. (8). Also, since the

leading-order term S

must be positive because the isother-

mal compressibility is positive [cf. (7)], classic hard-sphere

liquids are not hyperuniform. Figure 2 shows g

(r) and the

corresponding structure factor S(k) in three dimensions at

φ = 0.35 as obtained from the PY approximation. Note that the

Percus-Yevick approximations for 2D and 3D systems do not

exhibit structural precursors to the respective freezing transi-

tions, as manifested by “shoulders” in the second peak of g

(r)

observed in computer simulations.

It is noteworthy that one can create “stealthy” hyper-

uniform hard-sphere packings

by decorating points derived

from stealthy equilibrium conﬁgurations

with nonoverlap-

ping spheres.

B. Nonequilibrium disordered packings

Here we brieﬂy discuss three different nonequilib-

rium sphere packings: random sequential addition (RSA),

“ghost” random sequential addition, and random close packing

(RCP).

1. Random sequential addition

Perhaps one of the most well-known nonequilibrium

packing models is the random sequential addition (RSA)

packing procedure, which is a time-dependent process that

generates disordered sphere packings in R

; see Refs. 89–97.

The RSA packing process in the ﬁrst three space dimen-

sions has been used to model a variety of different con-

densed phases, including protein adsorption,

polymer oxi-

dation,

particles in cell membranes,

and ion implantation

in semiconductors.

In its simplest rendition, an RSA sphere packing is pro-

duced by randomly, irreversibly, and sequentially placing

nonoverlapping objects into a large volume in R

that at some

initial time is empty of spheres. If an attempt to add a sphere at

some time t results in an overlap with an existing sphere in the

packing, the attempt is rejected and further attempts are made

until it can be added without overlapping existing spheres. One

can stop the addition process at any ﬁnite time t, obtaining

an RSA conﬁguration with a time-dependent packing frac-

tion φ(t), but this value cannot exceed the maximal saturation

packing fraction φ

= φ(∞) that occurs in the inﬁnite-time

and thermodynamic limits. Even at the saturation state, the

spheres are never in contact and hence are not jammed in the

sense described in Sec. III C; moreover, the pair correlation

function g

(r) decays to unity for large r super-exponentially

fast.

100

The latter property implies that the structure factor

S(k) of RSA packings in R

must be an even and analytic

function at k = 0,

as indicated in Eq. (8). It is notable that

saturated RSA packings are not hyperuniform for any ﬁnite

space dimension.

96,97

In the one-dimensional case, also known as the “car-

parking” problem, the saturation packing fraction was obtained

analytically: φ

= 0.747 597 920 2. . ..

However, for d ≥ 2,

in the thermodynamic limit has only been estimated via

numerical simulations. The most precise numerical study to

date

has yielded φ

= 0.547 073 5 ± 0.000 002 8 for d = 2

and φ

= 0.384 130 7 ± 0.000 002 1 for d = 3. Estimates of φ

in higher dimensions are discussed in Sec. IV. RSA packings

have also been examined for spheres with a size distribution

101

and other particle shapes, including squares,

92,102,103

rectan-

gles,

104

ellipses,

103

superdisks,

105

and polygons

106,107

in R

and spheroids,

108

spherocylinders,

103

and cubes

109,110

in R

2. “Ghost” random sequential addition

There is a generalization of the aforementioned stan-

dard RSA process, which can be viewed as a “thinning”

process of a Poisson distribution of sphere and is param-

eterized by the positive constant κ that lies in the closed

interval [0, 1]; see Ref. 111. When κ = 0, one recovers the

standard RSA process, and when κ = 1, one obtains the “ghost”

FIG. 2. Pair statistics for an equilibrium hard-sphere

ﬂuid in three dimensions at φ = 0.35 as obtained from the

PY approximation.

2,12

Left panel: Pair correlation func-

tion g

(r) versus r/D, where D is the sphere diameter.

Right panel: The corresponding structure factor S(k) as

a function of the dimensionless wavenumber kD, which

clearly shows nonhyperuniformity.

020901-6 S. Torquato J. Chem. Phys. 149, 020901 (2018)

FIG. 3. Examples of two nonequilibrium packing mod-

els in two dimensions under periodic boundary condi-

tions. Left panel: A conﬁguration of the standard RSA

packing at saturation with φ

≈ 0.5. Right panel: A con-

ﬁguration of a ghost RSA packing at a packing fraction φ

very near its maximal value of 0.25. Note that the packing

is clearly unsaturated (as deﬁned in Sec. II) and there are

no contacting particles.

random sequential addition (RSA) process that enables one to

obtain exactly not only the time-dependent packing fractions

but all of the n-particle correlation functions g

for any n and d.

The reader is referred to Ref. 111 for details about the general

model.

In the ghost RSA process, one attempts to place spheres

into an initially empty region of space randomly, irreversibly,

and sequentially. However, here one keeps track of any rejected

sphere, which is called a “ghost” sphere. No additional spheres

can be added whenever they overlap an existing sphere or a

ghost sphere. The packing fraction at time t for spheres of

unit diameter is given by φ(t) =



1 − exp

(

−v

(1)t

)



, where

(R) is the volume of sphere of radius R. Thus, we see that as

t → +∞, φ = 2

−d

, which is appreciably smaller than the RSA

saturation packing fraction φ

in low dimensions; see Fig. 3

for 2D examples. Nonetheless, it is notable that the ghost RSA

process is the only hard-sphere packing model that is exactly

solvable for any dimension d and all realizable densities, which

has implications for high-dimensional packings, as discussed

in Sec. IV.

3. Random close packing

The “random close packing” (RCP) notion was pioneered

by Bernal

3–5

to model the structure of liquids and has been one

of the more persistent themes with a venerable history.

112–119

Two decades ago, the prevailing notion of the RCP state was

that it is the maximum density that a large random collection

of congruent (identical) spheres can attain and that this density

is a well-deﬁned quantity. This traditional view has been sum-

marized as follows: “Ball bearings and similar objects have

been shaken, settled in oil, stuck with paint, kneaded inside

rubber balloons–and all with no better result than (a packing

fraction of) . . .0.636”; see Ref. 113.

Torquato, Truskett, and Debenedetti

have argued that

this RCP-state concept is actually ill-deﬁned because “ran-

domness” and “closed-packed” were never deﬁned and, even

if they were, are at odds with one another. Using the

Lubachevsky-Stillinger (LS)

120

molecular-dynamics growth

algorithm to generate jammed packings, it was shown

that

fastest particle growth rates generated the most disordered

sphere packings with φ ≈ 0.64, but that by slowing the growth

rates larger packing fractions could be continuously achieved

up to the densest value φ

max

= 0.740 48. . . such that the

degree of order increased monotonically with φ. These results

demonstrated that one can increase the packing fraction by

an arbitrarily small amount at the expense of correspondingly

small increases in order, and thus, the notion of RCP is ill-

deﬁned as the highest possible density that a random sphere

packing can attain. To remedy these ﬂaws, Torquato, Truskett,

and Debenedetti

replaced the notion of “close packing”

with “jamming” categories (deﬁned precisely in Sec. III C)

and introduced the notion of an “order metric” to quantify

the degree of order (or disorder) of a single packing con-

ﬁguration. This led them to supplant the concept of RCP

with the maximally random jammed (MRJ) state, which is

deﬁned, roughly speaking, to be that jammed state with a min-

imal value of an order metric (see Sec. III C 4 for details).

This work pointed the way toward a quantitative means of

characterizing all packings, namely, the geometric-structure

approach.

We note that whereas the LS packing protocol with a

fast growth rate typically leads to disordered jammed states

in three dimensions, it invariably produces highly crystalline

“collectively” jammed packings in two dimensions. Figure 4

vividly illustrates the differences between the textures pro-

duced in three and in two dimensions (see Sec. III B for further

remarks).

C. Geometric-structure approach

to jammed packings

A “jammed” packing is the one in which each particle is

in contact with its nearest neighbors such that the mechani-

cal stability of a speciﬁc type is conferred to the packing, as

detailed below. Two conceptual approaches for their study have

emerged. One is the “ensemble” approach,

3,5,27,50–52,122–129

which for a given packing protocol aims to understand

typical conﬁgurations and their frequency of occurrence.

The other more recent one is the “geometric-structure”

approach,

49,56,121,130–134

which emphasizes quantitative char-

acterization of single-packing conﬁgurations without regard

to their occurrence frequency in the protocol used to produce

them. Here we focus on the latter approach, which enables

one to enumerate and classify packings with a diversity of

order/disorder and packing fractions, including extremal pack-

ings, such as the densest sphere packing and MRJ packings.

1. Jamming categories

Three broad and mathematically precise “jamming” cat-

egories of sphere packings can be distinguished depending on

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