Universal jamming phase diagram in the hard-sphere limit

We present a new formulation of the jamming phase diagram for a class of glass-forming fluids consisting of spheres interacting via finite-ranged repulsions at temperature $T$, packing fraction $\varphi$ or pressure $p$, and applied shear stress $\Sigma$. We argue that the natural choice of axes for the phase diagram are the dimensionless quantities $T/p\sigma {}^3$, $p\sigma {}^3/\epsilon$, and $\Sigma /p$, where $T$ is the temperature, $p$ is the pressure, $\Sigma$ is the stress, $\sigma$ is the sphere diameter, $\epsilon$ is the interaction energy scale, and $m$ is the sphere mass. We demonstrate that the phase diagram is universal at low $p\sigma {}^3/\epsilon$; at low pressure, observables such as the relaxation time are insensitive to details of the interaction potential and collapse onto the values for hard spheres, provided the observables are nondimensionalized by the pressure. We determine the shape of the jamming surface in the jamming phase diagram, organize previous results in relation to the jamming phase diagram, and discuss the significance of various limits.

Figure 1

(a) Shear stress $\Sigma$ vs strain rate $\mathrm{\gamma ̇}$ in standard molecular dynamics units for $\alpha =2$, two different values of $T/p\sigma {}^3$, and four different values of $p\sigma {}^3/\epsilon$. The filled symbols represent $T/p\sigma {}^3=0.03$ and the open symbols represent $T/p\sigma {}^3=0.1$. The different shapes and colors represent different pressures: pink diamonds are $p\sigma {}^3/\epsilon ={10}^{-2.5}$; blue triangles are $p\sigma {}^3/\epsilon ={10}^{-3}$; red squares are $p\sigma {}^3/\epsilon ={10}^{-3.5}$; and orange pentagons are $p\sigma {}^3/\epsilon ={10}^{-4}$. (b) Same data, made dimensionless by the pressure. For comparison, data for hard spheres are also shown as black circles. As for the soft spheres, filled symbols represent $T/p\sigma {}^3=0.03$ and open symbols represent $T/p\sigma {}^3=0.1$.

Figure 2

Approach to the hard-sphere limit. (a) Dimensionless shear stress $\Sigma /p$ vs pressure $p\sigma {}^3/\epsilon$ at fixed dimensionless temperature $T/p=0.1$ and dimensionless strain rate $\mathrm{\gamma ̇}/\sqrt{p}=0.1$ for three different exponents $\alpha$. (b) Dimensionless relaxation time $\tau p^{1/2}$ vs pressure $p\sigma {}^3/\epsilon$ for the same parameters $T/p=0.1$, $\mathrm{\gamma ̇}/\sqrt{p}=0.1$, and three different values of $\alpha$. In each plot, the horizontal line represents the value for hard spheres.

Figure 3

Hard-sphere results interpolated at prescribed values of the dimensionless temperature $T/p\sigma {}^3$. (a) Dimensionless shear viscosity $\eta \sqrt{\sigma /pm}$ vs dimensionless shear stress $\Sigma /p$. (b) Dimensionless relaxation time $\tau \sqrt{p\sigma /m}$ vs $\Sigma /p$. (c) Packing fraction $\varphi$ vs $\Sigma /p$. Each connected line is a different value of $T/p\sigma {}^3$ (from top to bottom): 0.02 (blue down triangles), 0.03 (red down triangles), 0.04 (black circles), 0.05 (blue circles), 0.06 (red circles), 0.07 (black squares), 0.08 (blue squares), 0.09 (red squares), 0.1 (black up triangles), 0.15 (blue up triangles), 0.2 (red up triangles), 0.25 (black pentagons), and 0.3 (blue pentagons).

Figure 4

Ratio of shear viscosity to relaxation time for hard spheres. As in Fig. 3, each connected line represents a different value of $T/p\sigma {}^3$. (a) Ratio of shear viscosity to relaxation time in standard hard-sphere units, $\eta \sigma {}^3/T\tau$ vs $\Sigma /p$. (b) Ratio of dimensionless shear viscosity $\eta \sqrt{\sigma /\mathit{pm}}$ to dimensionless relaxation time $\tau \sqrt{p\sigma /m}$. The values of $T/p\sigma {}^3$ and the symbols are the same as in Fig. 3. In order from top to bottom in panel (a) and from bottom to top along the right side of panel (b), the values are 0.02 (blue down triangles), 0.03 (red down triangles), 0.04 (black circles), 0.05 (blue circles), 0.06 (red circles), 0.07 (black squares), 0.08 (blue squares), 0.09 (red squares), 0.1 (black up triangles), 0.15 (blue up triangles), 0.2 (red up triangles), 0.25 (black pentagons), and 0.3 (blue pentagons).

Figure 5

Jamming phase diagram for $\alpha =2$. (a) Jamming phase diagram in the equilibrium plane at $\Sigma /p=0$ spanned by $\{T/p\sigma {}^3,p\sigma {}^3/\epsilon \}$. (b) Jamming phase diagram in the universal plane at $p\sigma {}^3/\epsilon =0$ spanned by $\{T/p\sigma {}^3,\Sigma /p\}$. Panel (b) was constructed using hard spheres, which we showed to be equivalent to soft spheres at $p\sigma {}^3/\epsilon =0$. In panels (a) and (b), we show contours of equal dimensionless relaxation time spaced by half decades: $\tau \sqrt{p\sigma /m}={10}^{0.5}$ (green up triangles), $\tau \sqrt{p\sigma /m}=10$ (red diamonds), $\tau \sqrt{p\sigma /m}={10}^{1.5}$ (pink pentagons), $\tau \sqrt{p\sigma /m}={10}^2$ (blue squares), $\tau \sqrt{p\sigma /m}={10}^{2.5}$ (orange down triangles), and $\tau \sqrt{p\sigma /m}={10}^3$ (black circles). The contours are constructed by interpolation. (c) Full three-dimensional jamming phase diagram spanned by $\{T/p\sigma {}^3,\Sigma /p,p\sigma {}^3/\epsilon \}$ for $\alpha =2$. Shown are three sets of four contour lines. Each set is connected by a surface to guide the eye and represents a level set of dimensionless relaxation time. The three sets correspond to the logarithmically spaced dimensionless relaxation times (top to bottom): $\tau \sqrt{p\sigma /m}=10$ (red), $\tau \sqrt{p\sigma /m}={10}^2$ (blue), and $\tau \sqrt{p\sigma /m}={10}^3$ (black). Each set consists of a contour along four planes cut through the diagram: the equilibrium plane at $\Sigma /p=0$, the hard-sphere plane at $p\sigma {}^3/\epsilon =0$, a plane at $p\sigma {}^3/\epsilon =0.1$, and a plane at $p\sigma {}^3/\epsilon =0.2$. (d) Schematic illustration of various paths in the jamming phase diagram. See text for details.