A jamming plane of sphere packings

Significance

Jamming is a ubiquitous phenomenon that occurs when the viscosity diverges or the rigidity emerges in many soft matter systems, such as granular materials, glasses, foams, colloidal suspensions, emulsions, and polymers. Jamming of frictionless spheres is also closely related to the sphere packing and optimization problems. In practice, jammed packings can be obtained by various ways, among which, compression and shear are two widely employed protocols. Here, we demonstrate that compression and shear-jammed frictionless packings can be described under a unified framework called jamming-plane. Using computer simulations, we show that compression and shear jamming of frictionless spheres can occur at different densities in the limit of large systems, contrary to conventional wisdom, but the jamming criticality is protocol-independent.

Abstract

The concept of jamming has attracted great research interest due to its broad relevance in soft-matter, such as liquids, glasses, colloids, foams, and granular materials, and its deep connection to sphere packing and optimization problems. Here, we show that the domain of amorphous jammed states of frictionless spheres can be significantly extended, from the well-known jamming-point at a fixed density, to a jamming-plane that spans the density and shear strain axes. We explore the jamming-plane, via athermal and thermal simulations of compression and shear jamming, with initial equilibrium configurations prepared by an efficient swap algorithm. The jamming-plane can be divided into reversible-jamming and irreversible-jamming regimes, based on the reversibility of the route from the initial configuration to jamming. Our results suggest that the irreversible-jamming behavior reflects an escape from the metastable glass basin to which the initial configuration belongs to or the absence of such basins. All jammed states, either compression- or shear-jammed, are isostatic and exhibit jamming criticality of the same universality class. However, the anisotropy of contact networks nontrivially depends on the jamming density and strain. Among all state points on the jamming-plane, the jamming-point is a unique one with the minimum jamming density and the maximum randomness. For crystalline packings, the jamming-plane shrinks into a single shear jamming-line that is independent of initial configurations. Our study paves the way for solving the long-standing random close-packing problem and provides a more complete framework to understand jamming.

In three dimensions, the densest packing of equal-sized spheres has a face-centered cubic (FCC) or a hexagonal close-packing (HCP) structure, and the density (packing fraction) is ${\phi }_{\mathrm{F}\mathrm{C}\mathrm{C}}={\phi }_{\mathrm{H}\mathrm{C}\mathrm{P}}\simeq 0.74$. This was conjectured initially by the celebrated scientist Kepler in the 17th century, known as the Kepler conjecture, and was proved by the mathematician Hales about 400 y later (1).

The “random version” of the sphere-packing problem, however, remains unsolved. In the 1960s, based on the empirical observation that the packing fraction of ball bearings, when poured, shaken, or kneaded inside balloons, never exceeds a maximum value ${\phi }_{\mathrm{R}\mathrm{C}\mathrm{P}}\approx 0.64$, Bernal and Mason (2) introduced the concept of “random close packing” (RCP) to characterize the optimal way to pack spheres randomly. Although many experiments and simulations have reproduced random packings with a volume fraction around 0.64, an agreement on the exact value of ${\phi }_{\mathrm{R}\mathrm{C}\mathrm{P}}$ has not been reached. Torquato et al. (3) proposed that the idea of RCP should be replaced by a new notion called “maximally random jammed” (MRJ) state, where the randomness is measured by some order parameters that characterize the crystalline order. O’Hern et al. (4) designed a fast-quench protocol that generates randomly jammed isotropic packings of monodisperse spheres at ${\phi }_{\mathrm{J}}=0.639\pm 0.001$ in the thermodynamic limit, which is referred to as the jamming-point (J-point) (5). Later, based on mean-field calculations, Zamponi and coworkers predicted that the jamming density of amorphous packings should span over a range on the jamming-line (J-line) (6, 7), which has been supported in a number of numerical simulations (8–11).

Spheres can be constrained not only by compression but also by shear. Many experiments and simulations have reported shear jamming (SJ) in granular matter and suspensions (12–20), and it was suggested by several studies that frictional interactions are essential for SJ (21, 22). Different from compression jamming, the contact network of a shear-jammed packing is generally anisotropic.

1) In this study, we propose, and construct through numerical simulations, a jamming-plane (J-plane), which extends the domain of frictionless jammed states from the previously established J-point (4) and J-line (6), by including anisotropic states realized by SJ. The J-plane provides a framework to clarify the answers to the following two important questions (1 and 2).

Do isotropic jamming (IJ) and SJ occur at the same density in frictionless spheres, in the thermodynamic limit? Previously, several simulation studies (21, 22) showed that, for an infinitely large, unjammed, frictionless sphere system at a density $\phi$ below the IJ density ${\phi }_{\mathrm{J}}$, the probability to obtain jamming is zero at any applied shear strain, which leads to a conclusion that SJ necessarily occurs at the unique density ${\phi }_{\mathrm{J}}$ in the thermodynamic limit. In particular, the conclusion was based on finite-size analyses of data obtained from athermal quasistatic shear (AQS) simulations, where random configurations were used as initial conditions (21, 22). On the other hand, a recent mean-field theory (23) predicts the existence of SJ at different densities in deeply annealed, frictionless, thermal hard sphere (HS) glasses.

To clarify the issue, in this paper, we systematically examine the possibility of SJ of frictionless spheres, in strain-controlled (14, 21, 22) and stress-controlled (19) AQS simulations of soft spheres (SSs), as well as in strain-controlled thermal quasistatic shear (TQS) simulations of HSs (24, 25). Inspired by ref. 23, particular attention is paid to the impact of the glass transition on the routes to jamming (6). As a result, we show that frictionless SJ exists over a range of densities, when deeply supercooled liquid configurations (with crystallization suppressed), instead of purely random (or poorly annealed) configurations as in refs. 21 and 22, are used as the initial conditions. This is confirmed by all kinds of jamming protocols performed in this study, with finite-size analyses. We find that frictionless SJ could take place in unjammed sphere assemblies “denser” than the J-point density ${\phi }_{\mathrm{J}}$, which is the minimum possible jamming density. It means that if the shear is reversed, these systems will firstly unjam and then jam again but in the opposite direction, much as in the frictional case (12).

It should be noted that deeply annealed glasses are accessible by multiple protocols. Besides the thermal annealing approach used here and previously in (9–11), unjammed configurations above ${\phi }_{\mathrm{J}}$ can be also obtained by protocols like athermal cyclic overcompression (16) or athermal cyclic shear (17, 19). These mechanical training processes can be considered as effective annealing (19) and are reproducible in experiments. For example, it is possible to implement AQS using the newly developed multiring Couette shear set-up (18).

2) Static jamming can take place at various densities depending on compression protocols (6–11), while the dynamic jamming density, or the “critical state density” ${\phi }_{\mathrm{c}}$, obtained in stationary shear rheology, was shown to be unique (26–29). How can the two seemingly paradoxical observations be reconciled?

Under compression, the jamming density depends on the initial condition—the deeper the initial state is annealed, the higher jammed density is obtained. This “memory effect” is examined by detailed reversibility tests in this paper. It turns out that the states at ${\phi }_{\mathrm{J}}$ are irreversibly jammed and therefore “memoryless.”

On the other hand, under shear, even a deeply annealed system can be rejuvenated and eventually evolves into a steady state where the initial memory is completely lost. In the quasistatic and thermodynamic limits, the viscosity of steady states diverges at ${\phi }_{\mathrm{c}}$ from below jamming, and the yield stress vanishes at ${\phi }_{\mathrm{c}}$ from above. It was shown in ref. 19 that the same equations of states are shared by the steady states under shear and the isotropically jammed states compressed from ${\phi }_{\mathrm{J}}$, and the two jamming densities are very close to each other, ${\phi }_{\mathrm{c}}\simeq {\phi }_{\mathrm{J}}$.

Here, after exploring both compression and shear-jammed states in a systematic and well-controlled way, we find that the density ${\phi }_{\mathrm{c}}\simeq {\phi }_{\mathrm{J}}$ of memoryless (under both compression and shear) jammed states sets a lower density bound for frictionless jamming. Higher jamming densities are obtained only if the initial memory can be kept; the initial condition (degree of annealing) is irrelevant anymore for memoryless states at ${\phi }_{\mathrm{c}}\simeq {\phi }_{\mathrm{J}}$.

The properties of new states, obtained by SJ, deserve to be analyzed in detail. For example, is SJ reversible upon reverting the route to jamming (25)? Are shear-jammed packings isostatic (i.e., the average contact number per particle is $z_{\mathrm{j}}=2d=6$ in $d=3$ dimensions) as in the IJ case (4, 30)? Do compression and shear-jammed packings exhibit critical properties of the same jamming universality class (31)? How do the anisotropy (13, 32) and the bond-orientational order (33) of the contact network change with the jamming strain? What is the difference on SJ between amorphous states and crystals? All of these questions will be answered in this paper.

Results

Construction of the J-Plane.

We simulate a thermal HS model and an athermal SS model, which have the same continuous diameter distribution but different interparticle interactions (SI Appendix, S1. Models). Typically, a critical jammed packing can be considered as a HS system with zero nearest-neighbor separations or, equivalently, a SS system with zero nearest-neighbor overlappings. Choosing these models has a major advantage: By using an efficient swap algorithm (34), it is possible to prepare equilibrium (liquid) HS configurations over an extremely wide range of equilibrium densities ${\phi }_{\mathrm{e}\mathrm{q}}$ (SI Appendix, S2. Preparation of Initial Configurations Using a Swap Algorithm). The timescale of the most deeply annealed states (i.e., at the largest ${\phi }_{\mathrm{e}\mathrm{q}}$) significantly exceeds previous limitations in simulations, becoming even comparable to experimental scales (35). This breakthrough opens a whole new realm of possibilities for exploring physics in previously inaccessible domains (19, 24, 25, 34–37), including IJ (11). In this study, we apply the same methodology to study SJ, showing that such a quantitative advance can conceptually change our understanding: SJ is in fact possible in “frictionless” spheres, if deeply supercooled liquid configurations (more specifically, when ${\phi }_{\mathrm{e}\mathrm{q}}$ is above a certain threshold ${\phi }_{\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{t}}$) are used as the initial conditions in the jamming protocols.

Without loss of generality, one can assume that any sphere packing is jammed at a “jamming density” ${\phi }_{\mathrm{j}}$ and a “jamming strain” ${\gamma }_{\mathrm{j}}$, with IJ being the special cases when ${\gamma }_{\mathrm{j}}=0$. Therefore, a strain-density (${\gamma }_{\mathrm{j}}-{\phi }_{\mathrm{j}}$) J-plane, as demonstrated schematically in Fig. 1, provides a parameter space that can include all packings jammed by compression and shear protocols. We emphasize that, in this study, ${\phi }_{\mathrm{j}}$ is used to represent the jamming density of any packings, while ${\phi }_{\mathrm{J}}$ is the unique J-point density, which in general satisfies ${\phi }_{\mathrm{J}}\le {\phi }_{\mathrm{j}}$. To map out the J-plane numerically (Fig. 2A), we employ four standard jamming protocols (SI Appendix, S3. Jamming Protocols): athermal rapid compression (ARC) (4), strain-controlled AQS (21, 22), thermal compression (TC) (38–40), and strain-controlled TQS (25). A further confirmation is obtained by stress-controlled AQS simulations (SI Appendix, S7. Stress-Controlled Athermal Quasistatic Shear). Below, we briefly describe main features of the J-plane.

Fig. 1.

Schematic J-plane. The J-plane $\left({\phi }_{\mathrm{j}},{\gamma }_{\mathrm{j}}\right)$ is the region where frictionless amorphous jammed configurations at jamming density ${\phi }_{\mathrm{j}}$ and jamming strain ${\gamma }_{\mathrm{j}}$ exist. The J-point and J-line known from the previous studies are included as its subset. IJ and SJ protocols bring the system, which is initially in a thermalized HS glass state at a given density ${\phi }_{\mathrm{e}\mathrm{q}}$ (an example is indicated by the cross mark on the ${\phi }_{\mathrm{e}\mathrm{q}}$-axis), to jammed states along the associated SJ-line ${\gamma }_{\mathrm{j}}\left({\phi }_{\mathrm{j}};{\phi }_{\mathrm{e}\mathrm{q}}\right)$, whose end points are $\left\{{\phi }_{\mathrm{I}\mathrm{J}}\left({\phi }_{\mathrm{e}\mathrm{q}}\right),0\right\}$ and $\left\{{\phi }_{\mathrm{S}\mathrm{J}}={\phi }_{\mathrm{J}},\infty \right\}$. The J-plane is divided into reversible-jamming and irreversible-jamming regimes. Reversible- and irreversible-jamming routes are demonstrated schematically on corresponding free-energy landscapes. The arrows indicate how the system evolves on the free-energy landscape in a cycle of compression/shear (unjammed $\to$ jammed $\to$ unjammed). The jamming routine is reversible if the system remains in the same metastable glass basin after the cycle. Each state in the reversible-jamming regime can be related reversibly to the initially thermalized HS liquid state at $\left({\phi }_{\mathrm{e}\mathrm{q}},0\right)$. The lighter color in this region represents the smaller corresponding ${\phi }_{\mathrm{e}\mathrm{q}}$. The reversible-jamming regime is upper-bounded by the yielding-jamming separation line of thermal HSs $\left\{{\phi }_{\mathrm{c}}\left({\phi }_{\mathrm{e}\mathrm{q}}\right),{\gamma }_{\mathrm{c}}\left({\phi }_{\mathrm{e}\mathrm{q}}\right)\right\}$ (gray line) and left-bounded by the state-following line (red line), whose end point is the state-following jamming point at $\left\{{\phi }_{\mathrm{J}}^{\mathrm{S}\mathrm{F}},0\right\}$. How the J-line and the J-plane are bounded from above is an open question. The relationship between the IJ density ${\phi }_{\mathrm{I}\mathrm{J}}$ and the initial equilibrium density ${\phi }_{\mathrm{e}\mathrm{q}}$ (SI Appendix, Figs. S5 and S9) are visualized by the connections between the ${\phi }_{\mathrm{e}\mathrm{q}}$-axis and the ${\phi }_{\mathrm{j}}$-axis, in particular, ${\phi }_{\mathrm{I}\mathrm{J}}\left(0\le {\phi }_{\mathrm{e}\mathrm{q}}\le {\phi }_{\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{t}}\right)={\phi }_{\mathrm{J}}$ and ${\phi }_{\mathrm{I}\mathrm{J}}\left({\phi }_{\mathrm{e}\mathrm{q}}={\phi }^{\mathrm{S}\mathrm{F}}\right)={\phi }_{\mathrm{J}}^{\mathrm{S}\mathrm{F}}$ (see Table 1 for the definitions and values of these characteristic densities).

Fig. 2.

Numerical J-plane. (A) Numerical data of the SJ-lines, for N = 8,000 spheres and a few different ${\phi }_{\mathrm{e}\mathrm{q}}$ (same data as in SI Appendix, Figs. S7 and S12). The filled and open symbols correspond to data obtained from thermal and athermal protocols, respectively. The star represents the state point $\left\{{\phi }_{\mathrm{j}}=0.68,{\gamma }_{\mathrm{j}}=0.09\right\}$ (for ${\phi }_{\mathrm{e}\mathrm{q}}=0.643$) examined by the stress-controlled protocol in SI Appendix, Fig. S15. (B) J-plane colored according to the RMSD ${\mathrm{\Delta }}_{\mathrm{r}}$ measured by one cycle of AQS (see SI Appendix, Fig. S13 for the detailed data of ${\mathrm{\Delta }}_{\mathrm{r}}$). The route to jamming is reversible/irreversible if ${\mathrm{\Delta }}_{\mathrm{r}}$ is below/above a threshold ${\mathrm{\Delta }}_{\mathrm{t}\mathrm{h}}=0.025$. The reversible-jamming (blue) and the irreversible-jamming (green) regimes are separated by the yielding-jamming separation line (pentagon line) and the state-following line corresponding to the thermal SJ-line for ${\phi }_{\mathrm{e}\mathrm{q}}={\phi }_{\mathrm{S}\mathrm{F}}$ (red filled triangle line). Data are averaged over 100 to 2,000 independent samples. Error bars represent the SE of the mean in this paper.

The J-plane is a collection of all state points $\left\{{\phi }_{\mathrm{j}},{\gamma }_{\mathrm{j}}\right\}$ representing amorphous jammed packings. Our simulation results are summarized in the numerical J-plane Fig. 2A and in SI Appendix, we explain in detail how they are obtained by using athermal (SI Appendix, S4. Exploring the Jamming-Plane Using Athermal Protocols) and thermal (SI Appendix, S5. Exploring the Jamming-Plane Using Thermal Protocols) jamming protocols. Note that the J-plane may quantitatively depend on model parameters such as the polydispersity, and it does not include partially ordered packings (3, 41), as crystallization is highly suppressed by the large polydispersity in our models (34). The J-plane contains the following three elements.

1) The J-point: The J-point (4) at $\left\{{\phi }_{\mathrm{J}},0\right\}$ is a special, unique point on the J-plane, because its density ${\phi }_{\mathrm{J}}$ is the lowest possible density among all state points. The packings at the J-point can be generated by ARC from random initial configurations with ${\phi }_{\mathrm{e}\mathrm{q}}=0$, as done in ref. 4 (SI Appendix, S4. Exploring the Jamming-Plane Using Athermal Protocols, A. Jamming-Point) or, more generally, for any initial configurations with ${\phi }_{\mathrm{e}\mathrm{q}}\le {\phi }_{\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{t}}$ (10), where ${\phi }_{\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{t}}\approx 0.56$ is the onset density of glassy dynamics (35) (SI Appendix, Fig. S5). For our model, the finite-size analysis (SI Appendix, Figs. S3 and S4) gives ${\phi }_{\mathrm{J}}=0.655\left(1\right)$ in the thermodynamic limit (see Table 1 for a summary of all relevant densities). The vertical line $\left\{{\phi }_{\mathrm{j}}={\phi }_{\mathrm{J}},{\gamma }_{\mathrm{j}}\right\}$ (the gray vertical line starting from the J-point in Fig. 1) sets the leftmost boundary of the J-plane: No packing exists below ${\phi }_{\mathrm{J}}$ in the thermodynamic limit. The packings shear-jammed below ${\phi }_{\mathrm{J}}$ are due to the finite-size effect (SI Appendix, Fig. S3), as previously noticed in refs. 21 and 22.

Table 1.

Summary of relevant densities

${\phi }_{\mathrm{d}}$ (36)	${\phi }_{\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{t}}$ (35)	${\phi }_{\mathrm{J}}$	${\phi }_{\mathrm{I}\mathrm{J}}^{\mathrm{m}\mathrm{i}\mathrm{n},\mathrm{t}\mathrm{h}}$	${\phi }^{\mathrm{S}\mathrm{F}}$	${\phi }_{\mathrm{J}}^{\mathrm{S}\mathrm{F}}$
0.594(1)	0.56	0.655(1)	0.665	0.60	0.67

The table summarizes the values of the dynamical glass transition crossover density ${\phi }_{\mathrm{d}}$, the onset density φ_onset of glassy dynamics, the J-point density ${\phi }_{\mathrm{J}}$ , the minimum IJ density ${\phi }_{\mathrm{I}\mathrm{J}}^{\mathrm{m}\mathrm{i}\mathrm{n},\mathrm{t}\mathrm{h}}$ obtained by the thermal protocol, the state-following density ${\phi }^{\mathrm{S}\mathrm{F}}$, and the state-following jamming density ${\phi }_{\mathrm{J}}^{\mathrm{S}\mathrm{F}}$.

2) J-line: The J-line $\left\{{\phi }_{\mathrm{I}\mathrm{J}},0\right\}$ (the blue line at the bottom of Fig. 1), with

$${\phi }_{\mathrm{I}\mathrm{J}}={\phi }_{\mathrm{I}\mathrm{J}}\left({\phi }_{\mathrm{e}\mathrm{q}}\right),$$ [1]

is formed by the state points of isotropic packings. The isotropic packings are obtained by isotropic compression protocols (ARC or TC), and the IJ density ${\phi }_{\mathrm{I}\mathrm{J}}$ depends on the density ${\phi }_{\mathrm{e}\mathrm{q}}$ of the initial equilibrium HS configuration before compression (see SI Appendix, S4.Exploring the Jamming-Plane Using Athermal Protocols, B. Jamming-Line: The Athermal Case and S5. Exploring the Jamming-Plane Using Thermal Protocols, C. Jamming-Line: The Thermal Case for how to obtain the J-line numerically) (9–11).

While the lower bound of J-line, ${\phi }_{\mathrm{I}\mathrm{J}}={\phi }_{\mathrm{J}}$ can be obtained by ARC, the minimum IJ density generated by TC is ${\phi }_{\mathrm{I}\mathrm{J}}^{\mathrm{m}\mathrm{i}\mathrm{n},\mathrm{t}\mathrm{h}}=0.665$, which is above ${\phi }_{\mathrm{J}}$ due to thermal activations (SI Appendix, S5. Exploring the Jamming-Plane Using Thermal Protocols, A. Minimum Isotropic Jamming Density).

According to the mean-field theory (6), the J-line Eq. 1 is bounded from above by the “glass close-packing” density, ${\phi }_{\mathrm{G}\mathrm{C}\mathrm{P}}={\phi }_{\mathrm{I}\mathrm{J}}\left({\phi }_{\mathrm{K}}\right)$, where ${\phi }_{\mathrm{K}}$ is the Kauzmann point density. However, reaching this point is beyond the current computational power.

In practice, the maximum IJ density ${\phi }_{\mathrm{I}\mathrm{J}}^{max}$ depends on the protocol efficiency. For example, athermal training protocols typically give ${\phi }_{\mathrm{I}\mathrm{J}}^{max}\approx {\phi }_{\mathrm{J}}+0.02$ (16, 17). In this study, we are able to reach ${\phi }_{\mathrm{I}\mathrm{J}}^{max}\approx {\phi }_{\mathrm{J}}+0.035$, thanks to the efficient swap algorithm (11).

For clarity, let us note again that there are other protocols to prepare initial configurations, equilibrated to certain extents, such as the cyclic compression (16) and the cyclic shear protocols (17). For them, ${\phi }_{\mathrm{I}\mathrm{J}}$ depends on parameters that are specific to the protocol. For example, in the cyclic compression protocol, ${\phi }_{\mathrm{I}\mathrm{J}}\left({\phi }^{max}\right)$ depends on the maximum over-compression density ${\phi }^{max}$ (16), and in the cyclic shear protocol, ${\phi }_{\mathrm{I}\mathrm{J}}\left({\gamma }^{max}\right)$ depends on the maximum shear strain ${\gamma }^{max}$ (17).

3) SJ-lines: Each SJ-line (e.g., the curved line connecting the yellow circle and the blue triangle in Fig. 1),

$${\gamma }_{\mathrm{j}}={\gamma }_{\mathrm{j}}\left({\phi }_{\mathrm{j}};{\phi }_{\mathrm{e}\mathrm{q}}\right),$$ [2]

or, equivalently,

$${\phi }_{\mathrm{j}}={\phi }_{\mathrm{j}}\left({\gamma }_{\mathrm{j}};{\phi }_{\mathrm{e}\mathrm{q}}\right),$$ [3]

represents the functional dependency between ${\gamma }_{\mathrm{j}}$ and ${\phi }_{\mathrm{j}}$ for a given ${\phi }_{\mathrm{e}\mathrm{q}}$. The states with a nonzero jamming strain ${\gamma }_{\mathrm{j}}$ are said to be “shear-jammed.” The lower end point of the SJ-line at ${\gamma }_{\mathrm{j}}=0$ is nothing but an IJ point at $\left\{{\phi }_{\mathrm{I}\mathrm{J}}\left({\phi }_{\mathrm{e}\mathrm{q}}\right),0\right\}$ on the J-line Eq. 1, which is identical to

$${\phi }_{\mathrm{I}\mathrm{J}}\left({\phi }_{\mathrm{e}\mathrm{q}}\right)={\phi }_{\mathrm{j}}\left(0;{\phi }_{\mathrm{e}\mathrm{q}}\right),$$ [4]

and the upper end point is at $\left\{{\phi }_{\mathrm{S}\mathrm{J}}={\phi }_{\mathrm{J}},{\gamma }_{\mathrm{j}}=\infty \right\}$, in the thermodynamic limit (SI Appendix, Fig. S6), where all SJ-lines meet. The J-plane contains infinite number of SJ-lines, but numerically we only use a few typical SJ-lines to represent the J-plane as shown in Fig. 2A. In SI Appendix, S4. Exploring the Jamming-Plane Using Athermal Protocols, C. Jamming-Plane: The Athermal Case and S5. Exploring the Jamming-Plane Using Thermal Protocols, D. Jamming-Plane: The Thermal Case, we discuss how to obtain SJ-lines from simulations. In particular, the finite-size analysis (SI Appendix, Fig. S3) shows a clear difference between the SJ-lines for ${\phi }_{\mathrm{e}\mathrm{q}}=0$ (or more generally ${\phi }_{\mathrm{e}\mathrm{q}}\le {\phi }_{\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{t}}$) and ${\phi }_{\mathrm{e}\mathrm{q}}>{\phi }_{\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{t}}$: Only in the former case, the SJ-line becomes vertical in the thermodynamic limit (21, 22); in the latter case, the SJ-line is not vertical in that limit, which means that SJ could occur at different densities.

While the SJ-lines in Fig. 2A are constructed by using strain-controlled protocols, they can be also obtained by stress-controlled protocols, which are commonly used as well to investigate the behavior of jammed systems under shearing (42, 43). In stress-controlled AQS of SSs (SI Appendix, S7. Stress-Controlled Athermal Quasistatic Shear), the onset of SJ is signaled by a steep increase of mechanical stress ${\mathrm{\Sigma }}_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}}$ at $\gamma \approx {\gamma }_{\mathrm{j}}$ under the constant volume condition (SI Appendix, Fig. S15). A similar approach has been recently used in ref. 19 to study SJ in frictionless spheres.

Reversible-Jamming and Irreversible-Jamming.

Depending on the reversibility of the route to jamming, the whole J-plane is divided into two regions (see Figs. 1 and 2B).

1) Reversible-jamming regime: This regime contains jammed states that can be generated by “reversible” routes, in the sense that the initial unjammed state at $\left({\phi }_{\mathrm{e}\mathrm{q}},0\right)$ and the final jammed state at $\left({\phi }_{\mathrm{j}},{\gamma }_{\mathrm{j}}\right)$ are in the same metastable glass basin (or metabasin), and therefore the initial memory is kept (see the illustration in Fig. 1). In this regime, the quench to jamming corresponds to the so-called “state-following” dynamics in structural (44, 45) and spin (46–50) glasses. The terminology “state-following” is used here to emphasize the deep connection between the final jammed state and the initial equilibrium state, as the former is followed from the latter within the same metabasin. Although the Gardner transition (31), which takes place before reaching jamming, induces splitting of the metabasin into many marginally stable subbasins, it does not make the state-following dynamics irreversible as demonstrated in ref. 25.

2) Irreversible-jamming regime: In this regime, the routes to jamming are irreversible, in the sense that the initial unjammed state and the final shear-jammed state are in different metabasins, or such a metabasin cannot be defined for the initial state (see the illustration in Fig. 1). The memory of the initial condition is partially or completely lost. Very importantly, the J-point belongs to this regime.

The reversibility is determined numerically by examining the difference between the initial (unjammed) configuration before applying the jamming protocol and the final (unjammed) configuration after reverting the jamming route (Fig. 2B), quantified by the relative mean square displacement (RMSD) ${\mathrm{\Delta }}_{\mathrm{r}}$ (see SI Appendix, S6. Reversibility for details). The two regimes are separated by two cross-over lines: a “state-following line” that separates state-following and nonstate-following dynamics (SI Appendix, S6. Reversibility, C. Connection to Quench Dynamics) and a “yielding-jamming separation line” formed by state points $\left\{{\phi }_{\mathrm{c}}\left({\phi }_{\mathrm{e}\mathrm{q}}\right),{\gamma }_{\mathrm{c}}\left({\phi }_{\mathrm{e}\mathrm{q}}\right)\right\}$, which separates shear yielding and SJ in thermal HSs (23, 25, 51) (SI Appendix, S6. Reversibility, D. Connection to Yielding of Hard Sphere Glasses). The end point of the state-following line is the “state-following jamming point” at $\left\{{\phi }_{\mathrm{J}}^{\mathrm{S}\mathrm{F}},0\right\}$, which is compression quenched from the “state-following density” ${\phi }_{\mathrm{S}\mathrm{F}}$, i.e., ${\phi }_{\mathrm{J}}^{\mathrm{S}\mathrm{F}}={\phi }_{\mathrm{I}\mathrm{J}}\left({\phi }_{\mathrm{e}\mathrm{q}}={\phi }_{\mathrm{S}\mathrm{F}}\right)$. For our model, ${\phi }_{\mathrm{J}}^{\mathrm{S}\mathrm{F}}\approx 0.67$ and ${\phi }_{\mathrm{S}\mathrm{F}}\approx 0.60$ (SI Appendix, S5. Exploring the Jamming-Plane Using Thermal Protocols, B. State-Following Jamming Density and Table 1). The state-following density ${\phi }_{\mathrm{S}\mathrm{F}}\approx 0.60$ is very close to the dynamical glass transition density ${\phi }_{\mathrm{d}}=0.594\left(1\right)$ (36), both of which correspond to cross-overs rather than sharp phase transitions in finite dimensions. According to the mean-field theory, $\left\{{\phi }_{\mathrm{c}}\left({\phi }_{\mathrm{e}\mathrm{q}}\right),{\gamma }_{\mathrm{c}}\left({\phi }_{\mathrm{e}\mathrm{q}}\right)\right\}$ is a critical point in large dimensions (23), but simulation results suggest that this criticality is absent in three dimensions (25). Overall, the reversible and irreversible regimes are separated by gradual cross-overs, rather than reversible–irreversible transitions (17, 52, 53).

It should be noted that the reversibility is a property of free-energy landscape associated to the jammed state. The route to a jammed state in the reversible-jamming/irreversible-jamming regime is expected to be reversible/irreversible under any jamming protocols. Indeed, numerical results (SI Appendix, Fig. S9) confirm that, in the reversible-jamming regime, the relationship between the IJ density ${\phi }_{\mathrm{I}\mathrm{J}}$ and the initial equilibrium density ${\phi }_{\mathrm{e}\mathrm{q}}$ is independent of the jamming protocol (athermal or thermal), as well as the compression rate used in the thermal protocol. However, not every protocol can explore the entire J-plane. In fact, the thermal protocols can only access mainly the reversible part as shown in Fig. 2 (see SI Appendix, S5. Exploring the Jamming-Plane Using Thermal Protocols for details). This is because, in order to obtain low density packings, one needs to apply rapid TC, but if the compression is too fast, the generated packings become hypostatic ($z_{\mathrm{j}}<2d$; SI Appendix, Fig. S8).

So far, we have discussed how to construct the J-plane and the reversibility of the routes to jamming. Below, we analyze in detail the properties of packings on the J-plane.

Isostaticity and Jamming Universality

Any state point $\left\{{\phi }_{\mathrm{j}},{\gamma }_{\mathrm{j}}\right\}$ on the J-plane corresponds to a critical jamming state, right at a jamming–unjamming transition (4). Nearest-neighbor particles are just in touch in a critical jammed packing. Keeping the strain ${\gamma }_{\mathrm{j}}$ unchanged, the packing becomes overjammed when it is compressed from ${\phi }_{\mathrm{j}}$ to $\phi ={\phi }_{\mathrm{j}}+\delta \phi$, where $\delta \phi >0$, and becomes unjammed when it is decompressed from ${\phi }_{\mathrm{j}}$ to $\phi ={\phi }_{\mathrm{j}}-\delta \phi$. Quite remarkably, as shown below, the isotropically compression-jammed and shear-jammed states are all isostatic and belong to the same universality class, i.e., the jamming critical exponents near ${\phi }_{\mathrm{j}}$, from both below and above jamming, are independent of ${\gamma }_{\mathrm{j}}$ (including the case ${\gamma }_{\mathrm{j}}=0$).

Let us first show that the coordination number at the jamming–unjamming transition satisfies the isostatic condition. We compress the SS packings athermally from $\left\{{\phi }_{\mathrm{j}},{\gamma }_{\mathrm{j}}\right\}$ to $\left\{\phi >{\phi }_{\mathrm{j}},{\gamma }_{\mathrm{j}}\right\}$, keeping the shear strain ${\gamma }_{\mathrm{j}}$ unchanged, and then measure the coordination number $z$ (without rattlers) as a function of $\phi$. Fig. 3A shows that the coordination number $z$ satisfies the isostatic condition $z=z_{\mathrm{j}}=6$, at the unjamming transition where the mechanical energy density vanishes, $e_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}}=E_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}}/N\to 0$, from above jamming. Consistently, Fig. 3C shows that the isostatic condition also holds at the jamming transition where the reduced entropic pressure diverges, $p_{\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}}\to \infty$, from below jamming in thermal HSs. Moreover, the isostatic condition is valid for any packing along the SJ-line, independent of ${\gamma }_{\mathrm{j}}$ (Fig. 3 A and C).

Fig. 3.

Isostaticity and universality of jamming. We show scalings (A and B) above jamming $\phi >{\phi }_{\mathrm{j}}$ in athermal SSs and (C and D) below jamming $\phi <{\phi }_{\mathrm{j}}$ in thermal HSs, for ${\phi }_{\mathrm{e}\mathrm{q}}=0.643$ and N = 8,000. Data in A and B are obtained from athermal compressions of SS packings from ${\phi }_{\mathrm{j}}$ to $\phi$, for a few different ${\phi }_{\mathrm{j}}$ along the SJ-line (blue open triangles in Fig. 2 and SI Appendix, Fig. S7). (A) Mechanical energy density $e_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}}$ versus coordination number $z$. (B) Mechanical pressure $P_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}}$ and the excess coordination number $z-z_{\mathrm{j}}$ as functions of $\phi -{\phi }_{\mathrm{j}}$. The data are consistent with scalings Eqs. 5 and 6 (lines). (C) Cumulative structure function $Z\left(r\right)$ of HSs below jamming, for a few different ${\phi }_{\mathrm{j}}$ along the SJ-line (blue filled triangles in Fig. 2 and SI Appendix, Fig. S12). The black line indicates the isostatic coordination number $z_{\mathrm{j}}=6$. (D) The data of $Z\left(r\right)$ are consistent with the critical jamming scaling Eq. 9 (line).

Second, we show that all packings above jamming ($\phi >{\phi }_{\mathrm{j}}$) follow the same set of scaling laws. Under athermal compressions, the mechanical pressure $P_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}}$, the mechanical energy $E_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}}$, and the coordination number $z$ all increase in SS packings. Fig. 3A shows that, for packings along the SJ-line with the same ${\phi }_{\mathrm{e}\mathrm{q}}=0.643$ but different ${\phi }_{\mathrm{j}}$ (blue open triangles in Fig. 2A), the data of energy density $e_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}}$ versus $z$ collapse onto the same master curve, which vanishes at $z_{\mathrm{j}}=6$. Furthermore, the following scalings, which are well known for isotropically jammed packings (4), are also satisfied in shear-jammed packings (Fig. 3B):

$$P_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}}\sim \phi -{\phi }_{\mathrm{j}},$$ [5]

and

$$z-z_{\mathrm{j}}={\left(\phi -{\phi }_{\mathrm{j}}\right)}^{1/2}.$$ [6]

Third, we examine the scaling behavior below jamming ($\phi <{\phi }_{\mathrm{j}}$). To do that, we compute the cumulative structure function $Z\left(r\right)$ of HS packings along the SJ-line for ${\phi }_{\mathrm{e}\mathrm{q}}=0.643$. The packings are compressed or sheared until the pressure reaches $p_{\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}}=10^{12}$. The cumulative structure function is defined as

$$Z\left(r\right)=\rho {\int }_0^{r}ds4\pi s^{2}g\left(s\right),$$ [7]

where $g\left(s\right)$ is the pair correlation function,

$$g\left(s\right)=\frac{1}{4\pi s^2\rho N}⟨\sum _{i\ne j}\delta \left(s-r_{ij}/D_{ij}\right)⟩,$$ [8]

with $\delta \left(x\right)$ being the delta function, $\rho$ the number density, $D_{ij}=\left(D_i+D_j\right)/2$ the average diameter, and $r_{ij}$ the interparticle distance. The cumulative structure function $Z\left(r\right)$ exhibits a plateau at $z_{\mathrm{j}}=6$ (Fig. 3C). The growth from this plateau satisfies the scaling

$$Z\left(r\right)-z_{\mathrm{j}}\sim {\left(r-1\right)}^{1-\alpha },$$ [9]

where $\alpha =0.41269$ (31). This scaling is predicted by the mean-field theory and has been verified numerically for isotropically jammed packings in finite dimensions (31). Here, we show that it also holds for shear-jammed packings.

Anisotropy of Contact Networks

We use the anisotropy parameter $R_{\mathrm{A}}$, which is based on the fabric tensor $\widehat{R}$, to quantify the anisotropy of contact networks in jammed packings. The fabric tensor is defined as

$$\widehat{R}=\frac{1}{N}\sum _{i\ne j}\frac{{\mathbf{r}}_{ij}}{\left|{\mathbf{r}}_{ij}\right|}\otimes \frac{{\mathbf{r}}_{ij}}{\left|{\mathbf{r}}_{ij}\right|},$$ [10]

where ${\mathbf{r}}_{ij}$ is the vector connecting two particles $i$ and $j$ that are in contact, and $\otimes$ denotes a vector outer product. The eigenvalues of $\widehat{R}$ are denoted by ${\lambda }_1$, ${\lambda }_2$, and ${\lambda }_3$, and the coordination number is related to the eigenvalues by $z_{\mathrm{j}}={\lambda }_1+{\lambda }_2+{\lambda }_3$. The fabric anisotropy parameter $R_{\mathrm{A}}$ is defined as the difference between the largest and the smallest eigenvalues, normalized by $z_{\mathrm{j}}$, $R_{\mathrm{A}}=\left({\lambda }_{max}-{\lambda }_{min}\right)/z_{\mathrm{j}}$ (14).

Another important quantity to characterize the anisotropy is the contact-angle probability distribution $P_{\theta }\left(\theta \right)$, where the angle $\theta$ is defined through the coordinate transformation ${\mathbf{r}}_{ij}=\left(r_{ij}\sin \theta \sin \varphi ,r_{ij}\cos \varphi ,r_{ij}\cos \theta \sin \varphi \right)$, considering that the shear strain is applied in the $x-z$ plane. From the lowest order Fourier expansion, $P_{\theta }\left(\theta \right)$ is related to the fabric anisotropy parameter $R_{\mathrm{A}}$ via (32),

$$P_{\theta }\left(\theta \right)\approx \frac{1}{2\pi }\left[1+2R_{\mathrm{A}}\cos 2\left(\theta -{\theta }_{\mathrm{c}}\right)\right],$$ [11]

where ${\theta }_{\mathrm{c}}$ is the principle direction.

Apparently, for IJ (${\gamma }_{\mathrm{j}}=0$), the fabric anisotropy parameter $R_{\mathrm{A}}$ should be nearly zero, and the contact-angle distribution should be uniform, as confirmed in Fig. 4 A and C. For SJ, let us consider two cases. In the case of ${\phi }_{\mathrm{e}\mathrm{q}}=0$, the fabric anisotropy parameter immediately jumps to a finite value $R_{\mathrm{A}}\approx 0.03$ for nonzero ${\gamma }_{\mathrm{j}}$ and stays as a constant for larger ${\gamma }_{\mathrm{j}}$ (Fig. 4 A and B), which is consistent with the observation in ref. 54 (ref. 54 also suggests that $R_{\mathrm{A}}\left({\gamma }_{\mathrm{j}}\right)$ would jump discontinuously at ${\gamma }_{\mathrm{j}}=0$ in the thermodynamic limit). Accordingly, the contact-angle distribution $P_{\theta }\left(\theta \right)$, with ${\theta }_{\mathrm{c}}=135°$, also quickly converges to the asymptotic distribution (Fig. 4C). In the case of ${\phi }_{\mathrm{e}\mathrm{q}}=0.643$, which is above ${\phi }_{\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{t}}$, the fabric anisotropy does not change monotonically with ${\gamma }_{\mathrm{j}}$ (Fig. 4 A and B). At intermediate ${\gamma }_{\mathrm{j}}$, the $P_{\theta }\left(\theta \right)$ has a dumbbell shape, indicating a strong anisotropy (Fig. 4D). At larger ${\gamma }_{\mathrm{j}}$, both $R_{\mathrm{A}}$ and $P_{\theta }\left(\theta \right)$ converge to the same asymptotic behaviors as in the case of ${\phi }_{\mathrm{e}\mathrm{q}}=0$.

Fig. 4.

Contact anisotropy. Fabric anisotropy parameter $R_{\mathrm{A}}$ of packings along SJ-lines, as a function of jamming density ${\phi }_{\mathrm{j}}$ (A) and jamming strain ${\gamma }_{\mathrm{j}}$ (B). Data are obtained for N = 8,000 and a few different ${\phi }_{\mathrm{e}\mathrm{q}}$, by using the athermal protocol (filled symbols) and the thermal protocol (open symbols). Dashed lines in B represent data obtained for the critical state, i.e., steady states in AQS simulations under constant, vanishingly small pressures ($P_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}}=10^{-4}$ and $10^{-5}$, N = 8,000, averaged over 48 samples). (C) Contact-angle probability distribution $P_{\theta }\left(\theta \right)$ for ${\phi }_{\mathrm{e}\mathrm{q}}=0$ and ${\gamma }_{\mathrm{j}}=\mathrm{0,0.08},0.16,0.19,0.25$ (from left to right). (D) Contact-angle probability distribution $P_{\theta }\left(\theta \right)$ for ${\phi }_{\mathrm{e}\mathrm{q}}=0.643$ and ${\gamma }_{\mathrm{j}}=\mathrm{0,0.04},0.12,0.28,0.37$ (from left to right).

It is interesting to compare the anisotropy of jammed states on the J-plane to that of the critical state. Following ref. 19, steady states are obtained by constant pressure AQS of SSs at large shear strains ($\gamma \gg {\gamma }_{\mathrm{Y}}^{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}}\sim 0.1$), where sample-averaged physical quantities do not change anymore with the strain. The constant pressure simulation is realized by minimizing the enthalpy after each strain step (19). By definition, the critical state is a steady state in the zero pressure limit $P_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}}\to 0$. In Fig. 4B, we present $R_{\mathrm{A}}$ for two small pressures $P_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}}=10^{-4}$ and $10^{-5}$, where initial configurations are rapidly quenched from ${\phi }_{\mathrm{e}\mathrm{q}}=0$. At large $\gamma$, $R_{\mathrm{A}}$ (as well as other quantities such as the bond-orientational order parameter plotted in Fig. 5B) approaches a constant with a negligible pressure dependence, which means that the system has reached the critical state within the numerical precision. Special attention should be paid to 1) the generic compression-jammed state at the J-point $\left\{{\phi }_{\mathrm{j}}={\phi }_{\mathrm{J}}=0.655\left(1\right),{\gamma }_{\mathrm{j}}=0\right\}$, 2) the asymptotic shear-jammed state at $\left\{{\phi }_{\mathrm{j}}={\phi }_{\mathrm{J}},{\gamma }_{\mathrm{j}}\to \infty \right\}$, and 3) the critical state at ${\phi }_{\mathrm{c}}=0.656\left(1\right)$. We find that ${\phi }_{\mathrm{J}}={\phi }_{\mathrm{c}}$ within numerical precision, as has been already shown in ref. 19, and the asymptotic shear-jammed state and the critical state have a common, nonzero degree of anisotropy $R_{\mathrm{A}}\approx 0.03$, while the J-point state is clearly isotropic with $R_{\mathrm{A}}\approx 0$.

Fig. 5.

Order map. The weighted bond-orientational order parameter $Q_6^{\mathrm{w}}$ is plotted as a function of ${\phi }_{\mathrm{j}}$ (A) and ${\gamma }_{\mathrm{j}}$ (B), along SJ-lines with different ${\phi }_{\mathrm{e}\mathrm{q}}$, for N = 8000 systems. Dashed lines in B represent data obtained for the steady state in AQS simulations under constant, vanishingly small pressures (lines for $P_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}}=10^{-4}$ and $P_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}}=10^{-5}$ fall on top of each other).

Bond-Orientational Order

Although the crystalline order is absent in our polydisperse model, the packings could have structures at the local scale (11). We characterize such order by using the weighted bond-orientational order parameter (55), defined as

$$Q_{l,i}^{\mathrm{w}}=\sqrt{\frac{4\pi }{2l+1}\sum _{m=-l}^l{\left|\sum _{j=1}^{n_{\mathrm{b}}\left(i\right)}\frac{A_{ij}}{A_i}{Y}_{l,m}\left({\mathbf{r}}_{ij}\right)\right|}^2},$$ [12]

where $Y_{l,m}\left({\mathbf{r}}_{ij}\right)$ is the spherical harmonic of degree $l$ and order $m$, $A_{ij}$ is the area of the Voronoi cell face between particles $i$ and $j$, and $A_i={\sum }_j{A}_{ij}$. Here, we consider the average bond-orientational order parameter with $l=6$:

$$Q_6^{\mathrm{w}}=\frac{1}{N}\sum _i{Q}_{6,i}^{\mathrm{w}}.$$ [13]

Fig. 5A shows the $Q_6^{\mathrm{w}}-{\phi }_{\mathrm{j}}$ order map (3) obtained from the packings associated to the J-plane. The order parameter $Q_6^{\mathrm{w}}$ of isotropically jammed packings (${\gamma }_{\mathrm{j}}=0$) increases with ${\phi }_{\mathrm{e}\mathrm{q}}$. Ref. 11 has shown that this increase is inherited from the initial equilibrium configurations at ${\phi }_{\mathrm{e}\mathrm{q}}$. Along SJ-lines, the $Q_6^{\mathrm{w}}$ decreases with decreasing ${\phi }_{\mathrm{j}}$ or increasing ${\gamma }_{\mathrm{j}}$ (Fig. 5B). Interestingly, our result shows that the J-point has the minimum order $Q_6^{\mathrm{w}}$ and, therefore, the maximum randomness, among all state points on the J-plane. In this sense, the J-point coincides with the MRJ point introduced in ref. 3.

However, we emphasize that the crystalline order is excluded from our consideration, which is an essential difference from ref. 3. Furthermore, we find that the isotropic MRJ state (J-point state), the asymptotic shear-jammed state, and the critical state all display the same $Q_6^{\mathrm{w}}$ (Fig. 5).

Discussion

In this paper, the concept of J-plane is introduced and is realized numerically using athermal and thermal protocols. Thanks to the swap algorithm, we are able to explore the J-plane over a wide range of jamming densities. It is possible to replace the role of the swap algorithm by other protocols that are easier to be reproduced in experiments, such as the mechanical training protocols used in refs. 16 and 17. Indeed, similar SJ-lines as in SI Appendix, Fig. S7 have been obtained in ref. 16 (figure 6b in ref. 16), although within a much narrower range of densities. We therefore expect the J-plane to be reproducible in tapping and shear experiments of granular matter.

Our analysis reveals that the J-point is a rather special point on the J-plane. The state at the J-point has the minimum packing density and the maximum randomness among all possible amorphous, frictionless, jammed states. The remaining challenge to theories is to provide a first-principle understanding of the J-point. Our results show that ${\phi }_{\mathrm{J}}\ne {\phi }_{\mathrm{I}\mathrm{J}}\left({\phi }_{\mathrm{e}\mathrm{q}}={\phi }_{\mathrm{d}}\right)$, which rules out the possibility that the state at the J-point, is followed from the equilibrium state at the dynamical glass transition density ${\phi }_{\mathrm{d}}$. Instead, the recent spin-glass theory of quench dynamics (50, 56), together with earlier numerical studies (10), suggests that ${\phi }_{\mathrm{J}}={\phi }_{\mathrm{I}\mathrm{J}}\left({\phi }_{\mathrm{e}\mathrm{q}}\le {\phi }_{\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{t}}\right)$. Therefore, the generalization of the calculation in ref. 50 to sphere systems would be very appealing.

We show that the phase space of jammed states can be significantly extended by adding shear, which introduces anisotropy to the contact networks. Our results disprove the earlier understanding that SJ and IJ always occur at the same jamming density in the thermodynamic limit (21, 22). The reversibility of the routes to jamming has a deep connection to the reversibility of the corresponding thermal HS glasses upon quench or shear (25).

The J-plane proposed here shall not be confused with the zero-temperature Liu–Nagel jamming phase diagram (5). The latter is defined by a yield stress line, which separates jammed and unjammed regions on the stress-density plane (note that this stress refers to the mechanical stress of athermal SS systems, which is different from the entropic stress of HSs; SI Appendix, S1. Models), while the former is the collection of all jammed states on the jamming strain–jamming density plane, for which the mechanical stress is always zero. In principle, each state point $\left\{{\phi }_{\mathrm{j}},{\gamma }_{\mathrm{j}}\right\}$ on the J-plane can be extended into a Liu–Nagel-like phase diagram, as follows. A system at $\left\{{\phi }_{\mathrm{j}},{\gamma }_{\mathrm{j}}\right\}$ can be compressed into an overjammed state at $\left\{\phi >{\phi }_{\mathrm{j}},{\gamma }_{\mathrm{j}}\right\}$, whose mechanical yield stress is a function of $\phi$. This yield stress-density line gives a generalized Liu–Nagel phase diagram, with the original version (5) corresponding to the special case $\left\{{\phi }_{\mathrm{j}}={\phi }_{\mathrm{J}},{\gamma }_{\mathrm{j}}=0\right\}$. Indeed, very recently, such a generalization has been done for the cases $\left\{{\phi }_{\mathrm{j}}>{\phi }_{\mathrm{J}},{\gamma }_{\mathrm{j}}=0\right\}$, where the onset of yield stress becomes discontinuous at jamming (19).

Interestingly, frictionless SJ can be also observed in crystals such as FCC lattices (SI Appendix, S8. Shear Jamming of Face-Centered Cubic Crystals). Because crystals are in equilibrium, accordingly, the SJ-line becomes unique and independent of ${\phi }_{\mathrm{e}\mathrm{q}}$. The separation between reversible-jamming and irreversible-jamming, and its connection to yielding of thermal HSs, remains to be present (SI Appendix, Fig. S16).

Finally, our results should pave the way for a set of additional studies. Related open questions include, but are not limited to the following: Do the jammed states on the J-plane share the same rheological properties, before and after yielding (24, 25)? How can the J-plane be extended in order to integrate the effects of friction (13, 14, 57) and crystalline order (58)? Can we make a connection between reversible-jamming/irreversible-jamming discussed here and the reversible–irreversible transition in suspensions (52) and granular systems (17, 53)?

Materials and Methods

Full materials and methods are included in SI Appendix, where we describe in detail the simulation models (SI Appendix, S1. Models), the swap algorithm that is used to prepare initial configurations (SI Appendix, S2. Preparation of Initial Configurations Using a Swap Algorithm), and the protocols to obtain jammed configurations (SI Appendix, S3. Jamming Protocols and S7. Stress-Controlled Athermal Quasistatic Shear). We also explain how to explore the J-plane using athermal (SI Appendix, S4. Exploring the Jamming-Plane Using Athermal Protocols) and thermal (SI Appendix, S5. Exploring the Jamming-Plane Using Thermal Protocols) jamming protocols and how to analyze the reversibility of routes to jamming (SI Appendix, S6. Reversibility). We discuss shear jamming of FCC crystals in SI Appendix, S8. Shear Jamming of Face-Centered Cubic Crystals, and more general jamming protocols in SI Appendix, S9. More General Jamming Protocols.

Data Availability

All study data are included in the article and SI Appendix.