Mesoscopic properties of dense granular materials: An overview

Abstract

A granular material is a conglomeration of discrete solid particles. It is intrinsically athermal because its dynamics always occur far from equilibrium. In highly excited gaseous states, it can safely be assumed that only binary interactions occur and a number of kinetic theories have been successfully applied. However, for granular flows and solid-like states, the theory is still poorly understood because of the internally correlated structures, such as particle clusters and force networks. The current theory is that the mesoscale characteristics define the key differences between granular materials and homogeneous solid materials. Widespread interest in granular materials has arisen among physicists, and significant progress has been made, especially in understanding the jamming phase diagram and the characteristics of the jammed phase. In this paper, the underlying physics of the mesoscale structure is discussed in detail. A multiscale framework is then proposed for dense granular materials.

Keywords

granular matter / macroscopic structure / jamming phase transition

Introduction

Granular materials are important constituents in many industrial processes as well as in geophysical phenomena. The lower size limit for individual particles is approximately 0.1 mm and the upper size limit may be a few meters, e.g., boulders in debris flows. If the materials are dry or the viscous force exerted by the flowing fluid can be neglected, the forces between the particles are essentially only repulsive. Despite such simplicity, under external mechanical excitations, such as shearing, vibration or tapping, a granular material may exhibit complex behaviors, and may be roughly referred to as ordinary solids, liquids and gases. However, the physical characteristics are often substantially different from their ordinary counterparts. This has frequently led to the characterization of granular materials as a new state of matter. Great interest in understanding granular materials has arisen in the physics community in recent decades [1].

Because of its discrete nature, each particle has contact with a limited number of neighboring particles. Subjected to external loads, the transmission of force from one boundary to another can only occur via these inter-particle contacts [2]. Furthermore, a multiplicity of chains, along which force transmission occurs, may be constructed to establish a stable stress state. If the load changes, a more appropriate and efficient set of force chain clusters is instantly re-established. It is possible to make preliminary observations about the roles played by force chain clusters under different strain rates. Inside the assembly, the displacement of particles relative to each other is so slight that the displacement around the contact points is elastic. Thus, the spatial structure of force chain clusters is intrinsically recoverable, causing the granular material to behave like a solid. The underlying mechanism of granular plasticity is simple for low shear rates; it consists of topological events that rearrange particles, i.e., contacts opened and closed by relative sliding or rolling, which leads to an irreversible evolution of force chain clusters. When the strain increases, the yield stress is reached, beyond which the plastic irreversible evolutions of force chain clusters continue indefinitely, and the system then flows like a fluid. The key issue is to understand how the force chain clusters and the mechanical properties of the granular matter are related.

In the last decade, significant progress has been made in understanding the transition from liquid-like to solid-like states, or vice versa, which is known as non-equilibrium dynamical transition, also known as the jamming transition [3,4]. This study focuses on the features of the intrinsic processes at the mesoscale. In Section 2, the dynamic processes are analyzed and a generalized granular temperature is introduced to denote energy fluctuations. In Section 3, a multiscale framework is proposed. In Sections 4 through 6, advances in the mesoscopic properties near the jamming transition point are discussed in detail, including coordination number, elastic modulus, vibration density of states and pair-correlation function. In Section 7, some open problems are listed.

Dynamic processes at mesoscopic scale

If a granular material is dense, or if interstitial fluid effects can be neglected, the particles will play the greatest role in transport processes within the material. Each particle will contact a limited number of its neighbors and form force chains, or granular skeletons, to establish a stable stress state. Around the contact points, the resultant finite deformability can statically support only longitudinal compression, supporting only small transverse loads. If the load changes, some particles will jiggle and slide, i.e., they will briefly lose contact with one another, and a more appropriate and efficient set of force chains is re-established. This variation would occur rapidly, causing the associated particles to slide and rotate, increasing the corresponding kinetic energy.

The importance of the kinetic energy and potential energy of granular materials is well-known. Because k_BT is near zero, unless excited by external disturbances, each metastable configuration of the material will last indefinitely, and no thermal averaging over nearby configurations will occur. A granular system is not conservative. The particle surface friction and inelastic collisions will lead to dissipation of kinetic energy. For these reasons, granular matter is non-ergodic, and does not satisfy the fundamentals of statistical physics. It is believed that granular matter still obeys the fundamental principles of statistical mechanics, and some successes have been made over the years in their application. For example, Edwards’ generalization of statistical mechanics has been applied mostly within the context of compaction, where the main observable variable is volume [5].

From the perspective of matter structure, the simple fluids and ideal solids primarily involve two spatial scales, microscale and macroscale, which obey quantum mechanics and classical mechanics, respectively. Meanwhile, the macroscopic properties can be derived directly from the microscopic quantities by using statistical physics or kinetic theory; thus, the connection between the two scales is established. For a granular material, each particle is a solid and can be described with elastic or elasto-plastic mechanics. The collective behavior of all constituent particles can be either a solid state, fluid state or gaseous state. The key obstacle is that the macroscopic parameters of the granular material (e.g., friction, elasticity and plasticity) cannot be simply derived from the statistics of particle positions, velocities and the inter-particle forces; this challenges the application of statistical physics.

The form of the expression for the disordered motion of micro-molecules at the macro level is the thermodynamic entropy, or temperature, but the form of the expression for the mesoscale motions at the macro-level is still relatively unfamiliar and is a new research topic in physics. There is an important difference between disordered motion at mesoscale and the thermal motion of micro-molecules. Because the interaction between particles is normally inelastic, when the moving particles are not in a local equilibrium state, entropy is produced. Therefore, as long as the molecular motion of an ordinary gas reaches Maxwell’s velocity distribution, the gas is in thermodynamic equilibrium. There is no entropy production. When the moving granular system is excited to the Maxwell velocity distribution state, the kinetic energy carried by the particles will decrease, and this is always accompanied by the production of entropy. Entropy production is a process limited in time, which indicates that the disordered motion at the mesoscale cannot be attributed directly to the thermodynamic entropy. These meso-characteristics are the main differences between granular matter and ordinary elastic objects, causing them to exhibit significantly different mechanical behaviors.

The work performed on the system is partially dissipated into heat (corresponding to an increase in entropy density, s), while the remaining work is stored in particles as both elastic and kinetic energy. In rapid flows at large shear rates, the motion of a particle is believed to be composed of a mean component and a random component, analogous to the thermal motion of molecules. The random component is the granular temperature, T=\langle u\prime u\prime \rangle, where u' denotes the random velocity component. The disordered kinetic energy, w_g, is proportional to ρT_k, where ρ is the bulk density of the granular flow. The concept of granular temperature, or kinetic theory, also applies to dense granular gases in which no enduring contacts exist. The Enskog-Boltzmann equation has been used to obtain constitutive relations for moderately dense granular gases, and even to very dense granular gases [6].

From the characteristics of the force chain clusters transition, the current study proposes that the elastic energy can be further separated into ordered and disordered elastic energy. In a manner similar to that used for T_k, the fluctuations of elastic energy stored in particles that form force chain clusters is defined as T_e. T_e does not include the elastic energy fluctuations of rattlers, i.e., particles not engaged in the force network. To qualify as a rattler, a particle must have a coordination number smaller than (d + 1), where d is the system’s dimension. For rattlers, we believe that the fluctuations in energies are caused by transient contacts, and further contribute to the random motions of the particles. Thus, the fluctuations in the elastic energy of the rattlers have already been included in T_k. There are a few studies that attempt to define T_e, but they do not consider whether particles form force chains or act as rattlers (e.g., Ref. [7].). That is the major difference between this study and others [8,9]. In this study, a new generalized granular temperature is introduced as follows:

(1)T_{\mathrm{g}}=T_{\mathrm{k}}+T_{\mathrm{e}},

where T_g is the conjugate of s_g, such that s_{\mathrm{g}}\equiv \partial w/\partial T_{\mathrm{g}}, where w is the system’s energy density.

In granular statics, the grains are at rest, the granular skeleton is fixed, and all particles are unmoving; thus, T_g = 0, the value for which granular entropy is at its minimum value. It is analogous to absolute zero, which is the theoretical temperature where entropy is at its minimum value. In the classical interpretation, it is zero and the thermal energy of matter vanishes. For granular materials, because of surface friction and inelastic contacts between solid particles, the energy associated with the granular temperature, T_g, is always dissipated, and s_g will quickly dissipate into s, i.e., s_g→s. For example, computer simulations of spatially homogeneous systems of hard inelastic spheres show that the granular temperature decays with time as t^-2. To maintain T_g, energy must be continually added to the system to balance the energy lost from the dissipative collisions and surface friction. Thus, it can be argued that granular media are transiently elastic, i.e., the elastic stress relaxes to zero at a set rate. It is this irreversible relaxation that is perceived as plastic granular flow. For granular materials, the random velocities,<u′ >, are roughly proportional to the mean velocity gradient and consequently will be much smaller than the mean velocities of the particles. However, a clear definition of T_g has not yet been derived for quasi-static deformations and slow flows. There remain many open problems regarding the extent to which the various temperatures serve functions similar to their molecular counterparts.

A possible multiscale method

As shown in Fig. 1, the Boltzmann equation (BE) is the basis of the kinetic theory for granular gases or very dilute granular flows. One of the most important characteristics of a real granular gas is the existence of inelastic binary collisions. This feature of inelasticity alone is responsible for many of the qualitative differences between granular and normal gases. The transport coefficients can be obtained at the level of the Navier-Stokes equations. They are among the most important equations of non-equilibrium statistical mechanics, the area of statistical mechanics that deals with systems far from thermodynamic equilibrium. Rigorously, BE is not applicable for dense granular flows in which enduring contacts occur.

In dense granular systems, it is often found that the experimental values of stress-strain for granular materials exhibit scattering for series of samples with approximately the same initial properties. This indicates that the stress-strain relationships cannot be solely determined by initial averaged properties. Such uncertainty cannot be treated simply as randomness in the experiments, but should be treated as evolutions of the internal force chain clusters. Small differences in clusters evolve nonlinearly when the strain process is amplified. Therefore, the scale couplings are non-equilibrium, nonlinear and sensitive, and they have not yet been thoroughly studied. In other words, how do the physical properties of clusters transport to the assembly, and then couple with the assembly, and how can these effects be formulated? Some should be based on experiments, and some should be based on the complete analysis of mechanical behaviors. However, the theories on the mechanical properties of force chain clusters are far from well-developed. Although contact forces and arrangement of particles have clearly established a relationship between the stress and strain, they are not directly related to the dynamic processes from the elastic deformation to the instability and destruction of a granular system.

Jamming transition

There are a wide variety of disordered materials, including foams, gels, colloidal suspensions and granular materials. All these systems exhibit a common non-equilibrium transition from a fluid-like phase to a solid-like phase, the jamming transition, as shown in Fig. 2. Three parameters, temperature, T, volume fraction, ϕ, and shear stress, \sum, control the characteristics of jamming transition. Currently, physicists study the law of the jamming phase transition when one and/or two parameters change. For example, the normal phase diagram of glass transition is in the (1/ϕ)–T plane, and it is divided into a jammed state region (e.g., glass) and an unjammed state region (e.g., liquid) by a transition line. The phase diagram of a granular material, e.g., a foam or a gel, lies in the (\mathrm{1}/\varphi )-\sum plane, and the critical yield stress curve, \sum (\varphi ), divides the phase diagram into the jammed state region and the unjammed state region. For granular matter, the two states correspond to the granular solid and the granular liquid. In addition, there is another jamming phase transition in the T-\sum plane, although a realistic system has not yet been found that corresponds to it.

Figure 2(a) shows the jamming phase diagram of a granular assembly composed of non-cohesive particles [5]. The figure shows that the characteristics of the jamming phase transition depend on the path to reach the phase boundary. For an isotropic jammed granular assembly, when ϕ is reduced to a certain point along the ϕ axis, the bulk modulus and shear modulus becomes zero. This transition point is the J point, and the corresponding critical volume fraction is ϕ_c. The distance to the J point is defined as follows:

(2)\mathrm{\Delta }\varphi =\varphi -{\varphi }_{\mathrm{c}}.

Theoretical and numerical studies have found that granular systems composed of frictionless particles exhibit some critical behaviors near the J point, which resemble second order transition in some ways, e.g., the bulk modulus shows a power law scaling with Δϕ and there are some divergent characteristic lengths when Δϕ approaches zero. For the systems with finite size, the effect of the boundary on ϕ_c cannot be ignored, and ϕ_c is distributed in a set range; while for the systems with infinite size, ϕ_c is constant. In addition, the deformation of frictionless rigid particles is identically zero, and the granular systems always stay at the J point, i.e., Δϕ ≡ 0.

Figure 2(b) shows the jamming phase diagram of a granular system composed of cohesive particles [10]. The three control parameters are ϕ, k_BT/U and σ/σ₀, where U is the interaction energy between particles, σ is the shear stress, and σ₀ = k_BT/a³, a function of temperature, T, and particle radius, a, is the characteristic stress exerted on the system. As shown in the figure, a granular liquid may be transformed into a granular solid by either increasing ϕ, increasing U or decreasing σ. The curvature of the phase boundary curves in Fig. 2(b) differs significantly from those in Fig. 2(a), and the curves diverge at every corner. The maximum value of the 1/ϕ axis corresponds to the irreversible aggregation; the maximum value of the k_BT/U axis corresponds to the hard particle limit. The maximum value of the σ/σ₀ axis (i.e., strong attraction among particles and high volume fraction, ϕ) corresponds to a sintered solid. Furthermore, the characteristic yield stress diverges when 1/ϕ→0, while the system without cohesion between particles corresponds to a finite yield stress. Further study is needed to compare the behavior of these two types of jammed systems.

The jamming phase diagram of disordered materials can provide a unified description of the jamming phenomena, such as the colloid-glass transition. For granular matter, the fundamental characteristic of the jamming transition is the existence of a relatively stable internal force network. There are similar network structures in other disordered materials. However, when non-cohesive particles contact each other, there is a mutual repulsion between them, and it is more difficult to form force chain networks. In a system composed of cohesive particles, the network structure is stable and can be easily observed, enabling the direct study of the force chains. Currently, study of the characteristics at the J point continues to be one of the more challenging problems of soft condensed matter physics.

Frictionless soft sphere systems

With respect to theoretical analysis, systems of frictionless soft spheres are ideal to study the jamming transition. First, the J point of this system is well-defined. The jammed system has finite shear modulus and yield stress when the boundary pressure P>0; the shear modulus disappears when P→ 0, and thus the unjamming transition occurs. At this stage, the system corresponds to zero pressure, zero shear rate, and zero temperature in the jamming phase diagram, i.e., the J point. Secondly, the system is marginally stable at the J point, so the coordination number, Z, is close to the static value (i.e., Z = Z_iso). Thirdly, at a finite pressure, the mechanical and geometric properties of the jammed system present a peculiar power law scaling with Δϕ. The peculiar geometrical and mechanical properties of frictionless soft sphere systems in the vicinity of the J point will be introduced in the following.

Coordination number of the isostatic system

The coordination number, Z, is one of the key parameters used to describe the structure of granular packing. A system is mechanically stable only when Z is greater than or equal to a critical value. The simplest jammed system is an isostatic system, in which the boundary pressure is isotropic, the system is stable, and the contact force between particles can be determined by the conditions of force balance and torque balance. Compared to other jammed granular systems, Z in an isostatic system is small.

Z is dependent on the spatial dimension, d, of a granular system and the roughness of the particle surface. For a stable granular system composed of Nd-dimensional frictionless soft particles (e.g., N particles in a d-dimensional system), assuming that the total number of contact forces (the total degrees of freedom) is NZ/2, the number of force balance equations (the total number of constraints) is Nd. The contact force is thus solvable when Z≥2d, and Z_iso = 2d corresponds to the isostatic value. At the J point, the pressure of the system is zero, there is no particle deformation, and the jammed conditions require that the distance between the particles is exactly equal to the sum of their radii (just touching); therefore, the system has Nd positional degrees of freedom and NZ/2 position constraints. The system has a unique solution when Z≤2d, and Z_c = 2d corresponds to the critical coordination number of the J point. This analysis shows that the frictionless soft sphere system is isostatic at the J point, i.e., Z_c = Z_iso = 2d. Durian performed experiments shearing 2D liquid foam, in which the air bubbles can be regarded as soft particles, and the foam formed by the bubbles is a generalized 2D granular system [11]. The results indicated that Z_c = 4 at the J point. Brujic et al. measured the coordination number of 3D particles in an emulsion by using a laser scanning confocal microscope technique [12]; the results indicated that Z_c = 6. Durian also found that there was a scaling relation in 2D systems as follows:

(3)(Z-\mathrm{4})\sim {(\varphi -{\varphi }_{\mathrm{c}})}^{\mathrm{1}/\mathrm{2}},\mathrm{i}\mathrm{e}(\mathit{Z}\mathit{-}{\mathit{Z}}_{\mathrm{c}})=\mathrm{\Delta }Z\sim \mathrm{\Delta }{\varphi }^{\mathrm{1}/\mathrm{2}}

Subsequent studies have shown that this scaling relation is independent of the interaction potential and spatial dimension of the particles. In summary, Z has a well-defined value at the J point, and exhibits a peculiar power law relationship in the vicinity of the J point. Taking into account that the mechanical properties of granular systems are sensitive to Z, it can be expected that the elastic modulus will also show a similar scaling behavior in the vicinity of the J point.

Elastic modulus

The system composed of frictionless particles is a typical marginally connected solid. Its bulk modulus, B, and shear modulus, G, depend on the type of interaction potential, U, between particles. In general, there are three types of interaction potentials, as described below:

1) Hertzian interaction potential, U~δ^5/2 and k~P^1/3, where k is the particle stiffness coefficient, δ is the overlap between particles, and P is the boundary pressure.

2) Harmonic interaction potential, U~δ² and k is a constant; and

3) general power-law interaction potential, U ~ δ^α and k ~ δ^α-2~(Δϕ)^α-2.

Figure 3(a) shows B and G for 2D granular systems with Hertzian interaction potential between particles. The figure shows that B satisfies P^1/3 scaling. If the effective medium theory can be extended to the granular system, the shear modulus will scale as G ~ (Δϕ)^α-2, but the results of the numerical simulations show that the granular systems scale as G ~ (Δϕ)^α-3/2, and G~P^2/3 for Hertzian and Harmonic interaction potentials, respectively. The results in Fig. 3(b) show that G/B ~ ΔZ^1/3, where ΔZ ~ Δϕ^1/2 ~ P^1/3.

Vibrational density of states

Density of states is an important concept in condensed matter physics. The vibrational spectrum of density of states of granular systems reflects the information of the collective movement of particles; therefore, the vibrational mode and the related density of states are important means to study the abnormal behavior of the system in the vicinity of the J point. For a crystal, the low-frequency vibrational density of states satisfies the Debye relation as follows:

(4)D(\omega )~{\omega }^{d-\mathrm{1}},

where d is the spatial dimension (i.e., d = 3 for three-dimensional) and ω is the rotational frequency of vibration. If D(ω) deviates from the relation {\omega }^{d-\mathrm{1}}, the system exhibits anomalous behavior. O’Hern et al. [13] obtained the density of states spectrum of a 3D granular system with Harmonic interaction using a computer simulation, shown in Fig. 4. Apart from the J point (i.e., ϕ – ϕ_c is large), D(ω) tends to zero at low frequency, mainly consistent with the Debye scenario. When ϕ reduces to ϕ_c, D(ω) increases dramatically in the low frequency region, and a plateau appears. D(\omega )~{\omega }^{d-\mathrm{1}} is still valid at the left of the plateau, which is shown more clearly in Fig. 4(b). Generally, taking the frequency at the left side of the plateau as a characteristic frequency, further studies have shown that, approaching the J point, {\omega }^{*}becomes smaller, {\omega }^{*}~\mathrm{\Delta }Z, and \mathrm{\Delta }Z=\mathrm{0}, until {\omega }^{*}=\mathrm{0} at the J point. The behavior of the vibrational density of states D(ω) illustrates that the closer the state is to the J point, the greater the difference is between granular matter and an ordinary solid.

There is a simple explanation for the at low-frequency plateau in the density of states. For an isostatic system, e.g., jammed square (2D) or cubic (3D) with side length l, there are surface bonds on the order of l^d-1. And because the system is isostatic, cutting the bonds at the surface can create floppy zero-energy modes on the order of l^d-1 within the square or cube. The underlying floppy mode can cause the system to deform on a scale of l. The frequency is ω_l = O(1/l). Thus, N_l≈ l^d-1 modes can be produced in a box with V_l~ l^d, and the highest frequency of the mode is ω_l~l^-1, and the following is derived:

(5){\displaystyle {\int }_{\mathrm{0}}^{{\omega }_l}D(\omega )d\omega }\approx {\displaystyle \frac{N_l}{V_l}}\sim {\displaystyle \frac{\mathrm{1}}{l}}.

Assuming D(ω) scales as ω^α at low frequencies, then the following is also true:

(6){({\omega }_l)}^{\alpha +\mathrm{1}}~{\displaystyle \frac{\mathrm{1}}{l^{\alpha +\mathrm{1}}}}~{\displaystyle \frac{\mathrm{1}}{l}}.

Therefore, α = 0, i.e., D(ω) is constant at low frequencies.

Microscopic criterion for stability under compression

The boundary pressure can have an impact on the vibrational density of states. Assuming the contact length between two particles is s, the corresponding potential energy can be characterized by the normal stiffness and the tangential stiffness coefficients. If the normal relative displacement is δ and the normal stiffness coefficient is k, then the normal force is kδ. If the tangential relative displacement is x, the contact length increases by an amount proportional to x²/s and the work produced by the contact force ~-x^{\mathrm{2}}k\delta /s, which is proportional to the tangential stiffness ~kδ/s. When a plane wave propagates in disordered solids, the transverse and longitudinal components of the contacts are of the same order of magnitude, and the ratio of the two stiffness coefficients is ~δ/s, which is one of the measures of strain in solids. Therefore, the impact of compression on the plane wave in the small strain can be ignored. Taking into account that there is no relative longitudinal but only transverse displacement in the abnormal pattern that is produced by the soft mode, once the soft modes are deformed to generate abnormal patterns, they gain a longitudinal component on the order of \mathrm{\Delta }Z, while the transverse component remains on the an order of unity. Therefore, the relative correction in the energy of the abnormal pattern is on the order of -\delta /(s\mathrm{\Delta }{Z}^{\mathrm{2}}). The anomalous modes become unstable, and the system yields when the relative correction reaches the order of unity. As the state approaches the J point, \delta /s\propto \mathrm{\Delta }\varphi, and the stability condition becomes \mathrm{\Delta }Z>\mathrm{\Delta }{\varphi }^{\mathrm{1}/\mathrm{2}}. This inequality must be satisfied for all subsystems with size L>l*, and this inequality extends the Maxwell criterion to the case of finite compression. For the systems with smaller size, a violation of this inequality of fluctuation of the coordination numbers are allowed, e.g., the fluctuation of the coordination number can be scaled as \mathrm{\Delta }Z~{(\mathrm{\Delta }\phi )}^{\mathrm{1}/\mathrm{2}} in the vicinity of the J point.

Pair-correlation function

The pair-correlation function, g(r), describes how the particle number density varies as a function of the distance, r, from one particular particle. For a 2D granular system, g(r) is defined as follows:

(7)g(r)={\displaystyle \frac{dN}{\mathrm{2}\mathrm{\pi }dr}}{\displaystyle \frac{\mathrm{1}}{{\rho }_{\mathrm{0}}}},

where dN is the number of particles inside the area defined by the radius r ~ r + dr, and ρ₀ is the mean number density. In a 2D system, there are N particles distributed in the area S, thus ρ₀ = N/S.

The pair-correlation function, g(r), of the frictionless soft sphere system has many peculiar properties at the J point. It has been shown that a δ-function peak appears in g(r) at r = a for the monodisperse system composed of particles with diameter a. And on the high side of this δ–function, g(r) exhibits a power-law decay described by g(r)∝(r-a)^-0.5. This can be explained as the vestige of the marginal stability of the configurations existing before reaching ϕ_c. At the J point, the second peak of the pair-correlation function splits into two peaks, one at r=\sqrt{\mathrm{3}}a and the other at r = 2a.

Abate and Durian experimentally studied the pair-correlation function near the zero-temperature jamming transition using the upward flow of gas through the mesh to control the movement of ball bearings [14,15]. In those systems, the kinetic energy of the ball bearings increases monotonically with decreasing ϕ, and the kinetic energy becomes zero when ϕ→ϕ_c. It was found that the height of the first peak of the pair-correlation function, g₁, increases when approaching ϕ_c, and g₁ exhibited a local maximum at ϕ ≈ 0.74. The coordination number and geometrical characteristics of the Voronoi cell does not change for ϕ>0.74. It has been found that the second peak of the pair-correlation function for the case of nonzero kinetic energy may correspond to a thermal vestige of the height divergence of the first peak at the zero-temperature transition. Zhang et al. measured the pair-correlation function of a 2D bidisperse system of NIPA particles at nonzero temperatures [16], shown in Fig. 5. There is a maximum in g₁, and it can be explained as a thermal vestige of the jamming transition at zero temperature. The numerical simulation shows that g₁ of the repulsive soft sphere system is diverse during the jamming transition at zero temperature, as follows:

(8)g_{\mathrm{1}}\sim \mathrm{1}/(\varphi -{\varphi }_{\mathrm{c}}).

The system softens to a finite maximum at nonzero temperatures. The maximum height of g₁ decreases with the increase in temperature, showing that the maximum g₁ is a structural feature of the jamming transition, and it is apparent as a function of increasing density at fixed temperature, not as a function of temperature at fixed density or pressure.

Frictional soft sphere systems

The system of frictionless soft spheres has a well-defined jamming point. The system is isostatic at the jamming point (i.e., Z_c = Z_iso), and the change of the coordination number near the J point shows a hybrid of the first/second order characteristics, i.e., below the J point, Z = 0; at the J point, Z jumps from 0 to 2d; above the J point, (Z – Z_c) and the mechanical quantities of the jammed system exhibit a power law scaling with (ϕ – ϕ_c), which is independent of dimensions, interaction potential and polydispersity. In contrast with the frictionless soft sphere system, ϕ_c and Z_c of the frictional soft sphere system is not unique, i.e., both of them depend on the friction coefficient and the preparation history of the system. The granular system of frictional soft spheres is hyperstatic (i.e., Z_c>Z_iso) at the J point. Its mechanical properties scale with (Z-Z_{\mathrm{i}\mathrm{s}\mathrm{o}}^{\mu }), while (Z – Z_c) scales with (ϕ – ϕ_c). The hyperstatic quality of frictional soft sphere systems at the J point implies that the contact forces between particles cannot be uniquely determined by the geometry of the system.

Critical coordination number

The coordination number of frictional soft sphere systems at zero-pressure is the primary focus in this study. On the one hand, the just-touching condition is the same as that of frictionless soft sphere systems, i.e., the ZN/2 restriction is applied at the Nd particle position coordinates, and requires Z≤2d. On the other hand, there are ZNd/2 components of contact forces and Nd constraint equations of force and torque balance in a frictional system, which require Z≥d+ 1. Combining these two constraints, the range of the coordination number of the frictional soft sphere system is as follows:

(9)d+\mathrm{1}\le Z_{\mathrm{c}}\prec \mathrm{2}d.

The range of the coordination number based on the above constraint statistics include all the force configurations satisfying the balance condition of force and torque, but these configurations do not always satisfy the Coulomb criterion, f_t/f_n≤μ, where f_n and f_t are the normal force and tangential force components at the contact, respectively, and μ is the static friction coefficient. In the limit μ→∞, the Coulomb criterion is satisfied naturally, then the coordination number at the J point, Z, is close to the lower limit, as follows:

(10)Z_{\mathrm{i}\mathrm{s}\mathrm{o}}^{\mu }=d+\mathrm{1},

where Z_{\mathrm{i}\mathrm{s}\mathrm{o}}^{\mu } is the isostatic value of the frictional soft sphere system. For the case of finite μ, the Coulomb criterion must be considered, then the critical coordination number at the J point is a function of μ, i.e., Z_c(μ). Numerical simulations and experiments showed that when μ increases gradually from 0, Z_c does not jump from 2d to d + 1, but to Z_c. However, Z_c is not a clearly defined function. It depends on the preparation history of the granular system. Numerical simulations show that static systems with different Z and ϕ at the J point can be generated by changing the quenching rate. Therefore, not only the impact of Coulomb criterion but also the impact of preparation history must be considered when analyzing the coordination number of frictional systems at J points.

Generalized isostaticity

Numerical simulations show that some of the contacts lie at the Coulomb yield threshold in the slowly prepared systems, i.e., f_n and f_t are no longer independent and they satisfy f_t = μf_n,. The contacts where the normal contact force and tangential contact forces satisfy the above relation are fully mobilized contacts (FMCs); the number of FMCs will affect the microscopic criterion of stability. Assuming that the average number of FMCs per particle is n_m, then there are ZNd/2 contact force components, while there are n_mN fully mobilized contact constraints in addition to Nd force balance equations and d(d-1)N/2 torque balance constraint equations. Following the above discussion on the microscopic criterion for stability of smooth particles systems, the generalized microscopic criterion of stability can be introduced for frictional system as follows:

(11)Z\ge (d+\mathrm{1})+\mathrm{2}{n}_{\mathrm{m}}/d\equiv Z_{\mathrm{i}\mathrm{s}\mathrm{o}}^{\mathrm{m}}.

The system with n_{\mathrm{m}}={\displaystyle \frac{d}{\mathrm{2}}}(Z-Z_{\mathrm{i}\mathrm{s}\mathrm{o}}^{\mathrm{\mu }}) is an isostatic, or marginal, system, and the system with is a hyperstatic system. Usually, the system with the maximum number of fully mobilized contacts is called a generalized isostatic system, and it can be implemented by the slow preparation. Numerical simulations showed that the values of n_m and Z in the generalized isostatic line for P→0 satisfy the above constraints. The phase diagram for the slowly prepared system with varying μ is shown in Fig. 6. The figure shows that for the generalized isostatic line, the following applies:

(12)\mu \to \infty ,n_{\mathrm{m}}=\mathrm{0},Z\approx d+\mathrm{1},

(13)\mu \to \mathrm{0},Z\approx \mathrm{2}d,n_{\mathrm{m}}\approx d(d-\mathrm{1})/\mathrm{2}.

In generalized isostatic systems, a large number of FMCs in the system produces in almost zero energy deformations, resulting in a large number of low-energy excitations, and the density of states in the low frequency region increases greatly, which has been verified in the measurements of the vibrational density spectrum.

Z–ϕ phase diagram

The volume fraction, ϕ, at the J point is not unique for frictional systems, but depends on the friction coefficient and the preparation history. Song et al. established the statistical volume description of jammed states of frictional systems by proposing a partition function of volume ensembles [3], and obtained the Z-ϕ phase diagram describing the volume fraction of the system with different friction coefficients, as shown in Fig. 7. In terms of volume ensemble, random loose packing (RLP) and random close packing (RCP) correspond to the limits of compaction, X= ∞ and X = 0, respectively, and the density of the ground state of jammed granular matter is ϕ_RCP ≈ 0.634 and ϕ_RLP(Z) ≈Z/(Z+\mathrm{2}\sqrt{\mathrm{3}}),, where Z is a function of the friction coefficient. Figure 7 shows that the statistical interpretation of the RLP and RCP limits is the following:

1) The RCP limit describes the maximum ϕ of disordered packing. It uses the friction coefficient to characterize the ground state of hard sphere systems. As μ changes from 0 to ∞, the RCP state changes correspondingly, i.e., RCP in the phase diagram is not a single point.

2) The RLP limit describes the minimum ϕ for a given Z. Along the RLP lines, the ϕ value of the RLP decreases as μ increases from 0 to ∞.

3) Systems between the RLP and RCP limits have a finite X (i.e., the yellow area in Fig. 7).

Characteristic frequency of density of state and the modulus ratio G/K

A large number of experiments and numerical simulations show that Z deviates from Z_c in frictional soft sphere systems. Z – Z_c follows the square root scaling law with (ϕ – ϕ_c), i.e., Z – Z_c = Z₀(ϕ – ϕ_c)^1/2, where Z₀ is a fitting factor. This is analogous to (Z – Z₀)~P^1/3 for Hertzian contacts. Z-Z_{\mathrm{i}\mathrm{s}\mathrm{o}}^{\mathrm{\mu }} at finite pressure does not scale with ϕ – ϕ_c because Z_c of a frictional soft sphere system is different than the isostatic value Z_{\mathrm{i}\mathrm{s}\mathrm{o}}^{\mathrm{\mu }}=d+\mathrm{1}.

The calculation for the 2D vibrational density spectrum of frictional systems shows that the characteristic frequency, ω*, can be scaled linearly with (Z-Z_{\mathrm{i}\mathrm{s}\mathrm{o}}^{\mathrm{\mu }}), as shown in Fig. 8(a). Similarly, the study on shear modulus, G, and bulk modulus, K, of a 2D frictional system shows that G/K also scales with (Z-Z_{\mathrm{i}\mathrm{s}\mathrm{o}}^{\mathrm{\mu }}), as shown in Fig. 8(b). These findings suggest that the scaling law of frictional soft sphere systems is determined by the distance from the isostaticity, rather than the distance to the jamming.

Jamming of non-spherical particles

The previous results apply to static granular systems of soft spheres, combined with the practical situations such as irregular particle shape, non-static shear, etc. Some of the conclusions can be helpful in understanding jamming in other disordered systems, such as non-spherical particles.

Statistical arguments regarding degree of freedom imply that the coordination number of an ellipsoid at the J point is Z = Z_iso = d(d + 1). However, numerical and experimental results show that Z of slightly non-spherical ellipsoids at the J point is close to 2d, which suggests that a granular system of weak ellipsoidal particles has (N/2)(Z_iso – Z) soft modes. For spherical particles, Z→2d, the number of soft modes would be d(d – 1)N/2. For frictionless soft sphere systems, d(d – 1)N/2 soft modes can be obtained accurately, corresponding to free rotation spheres. The characteristics of soft modes in the non-spherical hypostatic systems under finite pressure can be studied by the vibrational density spectrum.

The jamming transition of 2D and 3D ellipsoid systems was carefully studied. Because each ellipsoid can be characterized by five degrees of freedom, the isostatic coordination number of ellipsoids is Z_{\mathrm{i}\mathrm{s}\mathrm{o}}^{\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{i}\mathrm{p}\mathrm{s}\mathrm{o}\mathrm{i}\mathrm{d}}=\mathrm{2}\times \mathrm{5}=\mathrm{10}. Numerical simulations showed that the average coordination number changes from Z_iso = 10 to 6 successively when the ellipticity ϵ = c/a (c and a are the semi-minor axis and the semi-major axis of the ellipse, respectively) changes from a very small value (i.e., the shape is far away from spherical) to ϵ = 1. Calculations show that: when Z<Z_iso, there are N_c(Z_iso – Z)/2 directions in the phase space of the particles. The particle coordinates can change along these directions without changing the interaction energy among the particles, which means that each static state configuration has many zero-frequency modes. The numerical calculation of the density of states of an ellipsoid particle system found that the number of zero modes is consistent with the theoretical expectation, namely the number of zero modes decreases with increasing δ_ϵ = ϵ – 1. Thus, with the increase of Z, more and more nonzero modes would appear. Zeravcic et al. found that there were two obvious separate sub-bands in the density of states for small ϵ (such as |δ_ϵ|<0.17), as shown in Fig. 9(a).The modes in the upper band of the density of states is very similar to the modes found in the spherical particle systems, and the start frequency, ω*, is a characteristic frequency, and can be scaled as ΔZ = Z – 6~|δ_ϵ|^1/2. Furthermore, when ϵ is small, the low-frequency band is composed mainly of the rotation modes, and its upper frequency scales linearly with |δ_ϵ|. When ϵ>0.17, these two bands merge into a mixed band, as shown in Fig. 9(b).

Even though the non-spherical system is hypostatic, the jamming of frictionless ellipsoid systems is very similar to jamming of frictional soft spherical systems. Near the jamming point, there are large numbers of soft modes that do not affect the stiffness of the system either for the frictional soft spherical system or the weak ellipsoid system.

Conclusions

The states of disordered materials can transform from the unjammed state to the jammed state, and the transition to jamming occurs through changing the thermodynamic quantities (i.e., temperature or density) and/or loads (i.e., the stress applied to the sample). Granular materials with repulsive contact interactions at zero temperature and zero shear stress provide a good sample for understanding the nature of jamming transition and jammed states. Large numbers of experimental and numerical studies have shown that seemingly simple granular matter (such as frictionless, spherical and deformable particles) exhibit many unusual geometrical and mechanical behaviors near the J point. To date, many studies have been conducted on the jamming transitions of granular systems, but the physical nature of jamming transition is still far from well-understood. The following problems remain open.

1) The concept of random loose packing and random close packing. Song et al. have applied the concept of volume ensemble to provide a statistical interpretation of the RCP and RLP limits of frictional soft sphere systems, i.e., the RCP and RLP limits correspond to the ground state density of jammed granular matter at the limit of X = 0 and X = ∞, respectively. There is still some controversy about the control and realization of X, requiring further study. Moreover, the volume fraction and average coordination number of a system not only depend on the friction coefficient but also on particle shape and the preparation history. Currently, there are many open questions regarding the concept of RCP and RLP of non-spherical systems. Is the limit of RCP and RLP the same in spherical systems? Why does the random spherical system have a smaller volume fraction than the random non-spherical system? These are a few of the relevant questions. To answer these fundamental questions, further studies including experiments and simulations are required, and new statistical theories must be developed.

2) The relation between the geometrical and mechanical properties. The frictionless soft sphere system is isostatic (i.e., Z_c = Z_iso) at the J point. The mechanical quantities and (Z – Z_c) exhibit scaling laws with (ϕ – ϕ_c), which are independent of the system dimensions, interaction potential and polydispersity. The frictional soft sphere system is hyperstatic (i.e., Z_c>Z_iso), and the mechanical properties scale with (Z-Z_{\mathrm{i}\mathrm{s}\mathrm{o}}^{\mathrm{\mu }}), while (Z – Z_c) scales with (ϕ – ϕ_c), implying that the mechanical properties are not uniquely determined by the geometry of the system. The study on the density of state of 3D spherical systems shows that the sub-band of weakly deformed spheroids corresponds to translational modes, and the characteristic frequency, ω*, can be scaled with (Z-Z_{\mathrm{i}\mathrm{s}\mathrm{o}}^{\mathrm{s}\mathrm{p}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}}) rather than (Z-Z_{\mathrm{i}\mathrm{s}\mathrm{o}}^{\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{i}\mathrm{p}\mathrm{s}}), then the change from sphere to weakly deformed ellipsoid can be seen as a smooth perturbation. Under large boundary pressures, the translational band and the rotational band merge into one mixed band, with characteristic frequency ω⁺, and which scales with (Z-Z_{\mathrm{i}\mathrm{s}\mathrm{o}}^{\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{i}\mathrm{p}\mathrm{s}}). There are (Z-Z_{\mathrm{i}\mathrm{s}\mathrm{o}}^{\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{i}\mathrm{p}\mathrm{s}})/\mathrm{2} soft modes in the system with . At the spherical shape limit, these modes correspond to local rotations, and they are delocalized when far from the spherical shape. In short, these systems seem simple, but exhibit new physics. The close link between the geometrical and mechanical response needs further study.

3) Nonlinear behaviors and fluctuations near jamming. To date, most studies have focused on the average quantities and linear responses of disordered materials. The mechanical responses are affine mainly far away from the J point, while the mechanical responses of disordered systems become non-affine near the J point. Moreover, the characteristics of different configurations in the finite-sized systems, such as coordination number and modulus, are also different. There are still some problems that we cannot answer clearly; e.g., How do we understand the fluctuations near jamming? What is the nature of the nonlinear yielding of the system near jamming? Similar problems in the system with finite shear and finite temperature need further study.

4) General picture of jamming. Jamming provides a framework for understanding the mechanics of disordered systems. The study on stable frictionless soft spheres shows that these systems exhibit rich spatial organization and unusual mechanical properties at the limit of isostatic jamming. An important task for the next few years is to expand the framework of jamming to more general systems, and take into account the impact of shear and temperature. However, how to extend the picture of frictionless soft spheres to the systems with general interactions and non-spherical shapes is still unknown. The physics appearing in the more general scenario of jamming is in need of further experimental and theoretical exploration.

References

[1] Jaeger H M, Nagel S E, Behringer R P. Granular solids, liquids, and gases. Reviews of Modern Physics, 1996, 68(4): 1259-1273

[2] Sun Q, Jin F, Liu J, Zhang G. Understanding force chains in dense granular materials. International Journal of Modern Physics B, 2010, 24(29): 5743-5759

[3] Song C, Wang P, Makse H A. A phase diagram for jammed matter. Nature, 2008, 453(7195): 629-632

[4] Liu A J, Nagel S R. Jamming is not just cool any more. Nature, 1998, 396(6706): 21-22

[5] Edwards S F, Oakeshott R B S. Theory of powders. Physica A, 1989, 157(3): 1080-1090

[6] Goldhirsch I. Introduction to granular temperature. Powder Technology, 2008, 182(2): 130-136

[7] Kondic L, Behringer R P. Elastic energy, fluctuations and temperature for granular materials. Europhysics Letters, 2004, 67(2): 205-211

[8] Sun Q, Song S, Jin F, Bi Z. Elastic energy and relaxation in triaxial compressions. Granular Matter, 2011, 13(6): 743-750

[9] Sun Q, Song S, Jin F, Jiang Y. Entropy productions in granular materials. Theoretical and Applied Mechanics Letters, 2012, 2(2): 021002

[10] Trappe V, Prasad V, Cipelletti L, Segre P N, Weitz D A. Jamming phase diagram for attractive particles. Nature, 2001, 411(6839): 772-775

[11] Durian D J. Foam mechanics at the bubble scale. Physical Review Letters, 1995, 75(26): 4780-4783

[12] Brujic J, Song C, Wang P, Briscoe C, Marty G, Makse H A. Impact of a projectile on a granular medium described by a collision model. Physical Review Letters, 2007, 98: 248001

[13] O'Hern C S, Silbert L E, Liu A J, Nagel S R. Jamming at zero temperature and zero applied stress: The epitome of disorder. Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 2003, 68(1): 011306

[14] Abate A R, Durian D J. Approach to jamming in an air-fluidized granular bed. Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 2006, 74(3): 031308

[15] Keys A S, Abate A R, Glotzer S C, Durian D J. Measurement of growing dynamical length scales and prediction of the jamming transition in a granular material. Nature Physics, 2007, 3(4): 260-264

[16] Zhang Z, Xu N, Chen D T N, Yunker P, Alsayed A M, Aptowicz K B, Habdas P, Liu A J, Nagel S R, Yodh A G. Thermal vestige of the zero-temperature jamming transition. Nature, 2009, 459(7244): 230-233

[17] van Hecke M. Jamming of soft particles: geometry, mechanics, scaling and isostaticity. Journal of Physics Condensed Matter, 2010, 22(3): 033101

[18] Somfai E., van Hecke M., Ellenbroek W. G., Shundyak K., van Saarloos W.Critical and noncritical jamming of frictional grains. Physical Review E, 2007, 75(2): 020301(R)

[19] Zeravcic Z, Xu N, Liu A J, Nagel S R, Van Saarloos W. Excitations of ellipsoid packings near jamming. Europhysics Letters, 2009, 87(2): 26001

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