The viscoelasticity of glass is an intriguing subject in rheology and condensed matter physics, and is of both theoretical and technological importance.^[1,2] Developing microscopic predictive theory for macroscopic rheological properties and mechanical responses remains a major challenge. In crystals, rigidity is attributed to the long-range density correlation.^[3] In amorphous materials, the glass transition occurs upon cooling (thermal-driven system) or densification (density-driven system), and is accompanied by the emergence or significant rise of shear elasticity.^[4] Such shear resistance cannot be interpreted as the consequence of long-range correlation, as glass lacks long-range order. Instead, it is related to the rigidity of the local cage where the motion of a particle is restricted by its neighbors.^[5] The rigidity of the cage, or the barrier for a particle to escape the cage, increases upon cooling or densification, leading to larger shear elasticity. Phenomenological and several microscopic theories have been constructed to predict the rheological and mechanical quantities.^[6–12] For instance, the phenomenological “soft glass rheology” (SGR) model is a well-known theoretical approach to predict the transport properties of soft amorphous materials. SGR is based on Bouchaud’s trap model, and has been developed to address the impact of strain in mesoscopic elements on the local barrier-hopping process. In the SGR model, the trapped energy is often supposed to have a harmonic form that lacks microscopic interpretation, and hence limits its quantitative and material-specific predictive ability.^[6–9] The first-principles microscopic mode-coupling theory (MCT) and dynamic mean field p-spin glass models have been extended to address the shear acceleration of glassy dynamics by de-correlating the memory function that dictates the cage constraint.^[10–12] However, the activated hopping process, which dominates the structural relaxation in the deeply supercooled regime, is missing in these theories.

Recently, considerable attention has been paid to two-dimensional (2D) colloidal glass. The 2D colloidal assembly can be realized experimentally by, for example, confining colloidal particles at the water–air interface.^[13–16] An approach based on the positional data of 2D colloids recorded by video microscope and simulations of hard disks has been developed to determine the shear modulus and elastic dispersion relations for 2D colloidal glass.^[16] The glass transition and elasticity of hard disks has also been studied by MCT.^[17] The localization lengths at the transition are similar in 2D and 3D colloids, correlating to the Lindemann melting criterion. Fuchs and coworkers have studied the elasticity of binary mixed 2D hard disks, and found that mixing different components causes the softening of the elastic modulus.^[18,19]

Based on structural information about the materials, the microscopic nonlinear Langevin equation (NLE) theory for supercooled liquid and glass can capture the physics of local caging and activated barrier-hopping dynamics.^[20–25] The NLE theory has already been successfully applied to the slow dynamics and mechanical response of hard-sphere colloidal glass and polymer glasses.^{[5,20,26–29]} In previous work, we extended NLE theory to quiescent 2D hard disks.^[30] Here, we further extend the theory to address the transport properties and mechanical response of glassy hard disks. On the basis of the Green–Kubo linear response theory^[31] and the modified binary collision mean field (BCMF) theory,^[32,33] the elastic shear modulus and viscosity can be formulated in terms of the static structure, dynamic localization length, and hopping time. The external load affects the viscoelastic properties by effectively altering the dynamic localization length and barrier, i.e., distorting the dynamic free energy or “the dynamic free e”. Quantitative comparisons of the dynamic characteristic length scales, barrier, transport coefficients, and relevant mechanical control variable relations between 2D and 3D colloidal suspensions are presented. The data for 3D hard-sphere colloids are taken from or obtained following the approach of Schweizer.^[28]

The remainder of this paper is organized as follows. In Section 2, we recall the basics of NLE theory for hard disks and introduce its generalization to include mechanical deformation. Section 3 describes the elastic shear modulus, viscosity, and alteration of the profile of dynamical free energy under mechanical deformation and the stress–strain relation. We conclude in Section 4.

To investigate the distortion of the dynamic free energy under different mechanical variables, we introduce the simplified nonlinear stress–strain relation γ = τ/G(τ), where γ is the dimensionless strain and G(τ) can be computed from Eq. (4). Figure 1 shows profiles of the dynamic free energy under stress (Eq. (9)). The influence of stress on the dynamic free energy reduces the barrier, shifts the localization lengths, and alters the well and barrier curvatures. The barrier is totally destroyed when the stress reaches a value known as the absolute yield stress τ_abs, i.e., under this stress, the glass is mechanically devitrified and the particles flow without any hopping process. Correspondingly, we can define the absolute yield strain γ_abs = τ_abs/G(τ_abs). The dependences of the barrier, localization lengths, and curvatures on the stress or strain are shown in Figs. 2 and 3. The barrier is destroyed when the strain exceeds 20%, which is higher than in the 3D analog. Dimensionless plots of barrier versus stress (inset of Fig. 2(a)) or strain (inset of Fig. 2(b)) show quasi-universal deductions, which are also observed in 3D. r_loc (r_b) increases (decreases) with increasing stress or strain (Fig. 3(a)). The curves of r_loc and r_b merge at the absolute yield point. Analogous behavior for the well and barrier curvature is apparent in Fig. 3(b). The well curvature controls the vibrational amplitude and frequency of the trapped particle. The vanishing of curvatures corresponds to the localized–delocalized transition of the particle motion.

Fig. 1.

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	Fig. 1. Dynamic free energy as a function of scalar displacement r at volume fraction ϕ = 0.709 for three typical shear stresses. The three characteristic lengths are marked in the inset.

Fig. 2.

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	Fig. 2. (a) Activated barrier as a function of shear stress (in units of kB T/σ2) at three volume fractions. The inset shows the same results over a reduced scale. (b) Activated barrier as a function of strain. The inset shows the same results over a reduced scale.

Fig. 3.

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	Fig. 3. (a) Localization length and location of barrier as functions of strain and stress (inset). Data for 3D hard spheres are shown as star symbols for comparison. (b) Well and barrier curvatures as functions of strain and stress (inset). K0 for 3D hard spheres shown as star symbols.

Figure 4(a) shows the increase in the elastic shear modulus upon densification. The shear modulus increases exponentially G′ = 1.6 × 10⁻³e^{16.6ϕ/ϕ_RCP} ∼ e^25.9ϕ (in units of k_BT/σ³) over the entire packing fraction range investigated. As a comparison, the modulus in 3D colloidal suspensions can be fitted by G′ = 4.172 × 10⁻⁷e^{23.0ϕ/ϕ_RCP} ∼ e^27.7ϕ. Plotted versus , a roughly linear relationship is obtained at high density. If we use the random close packing fraction ϕ_RCP as a normalizing factor, the moduli of 2D and 3D colloidal suspensions are nearly identical at high density. The random close packing fraction for 3D is ϕ_RCP = 0.64, and that for 2D is ϕ_RCP = 0.84. We suggest that ϕ_r = ϕ/ϕ_RCP is an appropriate variable for a comparison between different dimensions.^[30] The shear viscosity η is related to the elastic shear modulus and the alpha relaxation time. In Maxwell’s relation, we have η = G′τ_α. Figure 4(b) compares the viscosity obtained from Eq. (5) and Maxwell’s relation. The difference in the results is within an order of magnitude over the entire density range investigated. In this range (area fraction from ϕ_r = 0.75 to ϕ_r = 0.90), the viscosity of 2D hard disks increases more than seven orders of magnitude; the difference between Eq. (5) and the Maxwell relation is only large at intermediate area fractions and vanishes at low and high densities. The viscosity can be fitted by the exponential form (η/η₀)_2D = 19.88e^{92.39(ϕ/ϕ_RCP)36.7}. We find (η/η₀)_2D ∼18 and (η/η₀)_3D ∼10 at the crossover packing fraction. The kinetic fragility is given by m_2D = 43.8 and m_3D = 69 at ϕ/ϕ_RCP = 0.90625. When stress is applied, there is a significant decrease in viscosity, as expected (Fig. 4(b)). However, a sharper increase in viscosity (larger fragility) occurs at high density under stress.

Fig. 4.

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Fig. 4. Transport coefficients. (a) Quiescent elastic shear modulus as a function of the ratio and (inset), where ϕr is the packing fraction reduced by the random close packing fraction ϕRCP. (b) Shear viscosity and the product of elastic modulus and relaxation time as functions of reduced packing fraction. Data for 3D colloids are shown (line+symbol) for comparison. The viscosity under four different stresses is also shown. The inset shows the viscosity and the product in 3D in a wider range of (normal) packing fraction.

The stress–strain relation is shown in Fig. 5(a). Stress increases with strain until the absolute yielding strain γ_abs is reached. The values of γ_abs become smaller for denser systems. As in 3D colloids, the absolute yield stress increases exponentially with packing fraction. The dynamic yield stress, which is empirically defined as the stress under which the relaxation time reduces to some value such as τ_α/τ₀ = 10 or τ_α/τ₀ = 100, also increases exponentially with the packing fraction at high density (Fig. 5(b)). For τ_α/τ₀ = 100, τ_y increases more rapidly with the packing fraction. It seems likely that the two curves will merge at sufficiently high area fractions, but our model cannot reach this point. Note that the dynamic yield stress in 2D grows much more rapidly with respect to the packing fraction than in 3D hard spheres. Note also that τ_dyn increases from ∼0.6 to τ_dyn ≈ 10 when ϕ increases by approximately 1%. For comparison, in 3D hard spheres, ϕ must increase by about 5% to produce the same increment in τ_dyn.

Fig. 5.

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	Fig. 5. (a) Stress–strain relation for packing fractions of f = 0.844, 0.869 and 0.889. (b) Dynamic yield stress as a function of packing fraction for relaxation times of τα/τ0 = 10 and τα/τ0 = 100.

Figure 6 shows the softening of the elastic shear modulus under deformation. The modulus is normalized by its unperturbed value. The modulus decays faster with stress at lower packing fractions, whereas it decays faster with strain at higher packing fractions. At the same relative packing fraction ϕ_r, the modulus of 2D hard disks decays less rapidly with deformation than that of 3D hard spheres.

Fig. 6.

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	Fig. 6. Reduced elastic shear modulus as a function of (a) shear stress and (b) strain.

In the strain-rate-control deformation, the glass is under plastic flow, which follows the relation . One strain rate corresponds to a unique stress and viscosity. Figure 7 shows the typical shear thinning behavior. The reduction of viscosity happens at a smaller strain rate for denser systems (inset of Fig. 7). We nondimensionalize the strain rate using the characteristic stress relaxation time τ_stress = η (τ = 0)σ³/8k_BT, i.e., . The decay in viscosity obeys a power law at large strain rates. The power-law decay of viscosity spans a larger range of strain rate at higher density. We obtain Δ_2D = 0.667 for 2D hard disks, which is close to the 3D value Δ_3D = 0.69 for ϕ_r = 0.889. This value is consistent with the analytic prediction Δ = 2/3 in the p-spin glass model. The relation between the stress and strain rate is shown in Fig. 8. The flow stress increases monotonically with strain rate. Opposite trends can be observed in the stress for 2D and 3D colloids. At low density, the flow stress of 3D hard spheres is larger than that of 2D hard disks for the same reduced density and strain rate. On the contrary, at high density, the flow stress of 2D hard disks becomes larger.

Fig. 7.

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	Fig. 7. Shear viscosity as a function of strain rate. is the nondimensionalized strain rate.

Fig. 8.

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	Fig. 8. Relation between stress and strain rate.

The nonlinear Langevin equation theory has been extended to study the transport properties and mechanical response of 2D hard disks via the Green–Kubo formula and modified BCMF theory. Expressions for the elastic shear modulus and viscosity of hard disks were derived based on the physical picture of the thermal-activated barrier-hopping dynamics and the mechanically accelerated motion. The dynamic free energy profile is distorted under deformation. The barrier is reduced, and the positions and curvatures of the local well and the barrier shift.

On the basis of the distortion of dynamic free energy, we predict the softening of elasticity and the reduction of shear viscosity under deformation. These phenomena are qualitatively the same as 3D hard spheres. The difference is in the density dependence of viscosity or the product of elastic modulus and relaxation time. Quantitative differences are manifested by comparisons between 2D and 3D colloids based on the same reduced packing fraction. For example, the dynamic yield stress in 2D grows much more rapidly with respect to the packing fraction than in 3D; the flow stress of 2D colloids is smaller (larger) than that of 3D colloids at low (high) density.

In this article, our theory was applied to a spatially homogeneous system in which the mechanical deformation was assumed to be scalar. A static structure factor was supposed to be close to thermal equilibrium because of the slow loading of external force. This work provides a foundation for further generalization of the nonlinear Langevin equation theory to study slow dynamics and rheological behavior in binary or polydisperse mixtures of hard or soft disks and to address the thin film or finite confinement problem.