Testing mean-field theory for jamming of non-spherical particles: contact number, gap distribution, and vibrational density of states

Abstract

We perform numerical simulations of the jamming transition of non-spherical particles in two dimensions. In particular, we systematically investigate how the physical quantities at the jamming transition point behave when the shapes of the particle deviate slightly from the perfect disks. For efficient numerical simulation, we first derive an analytical expression of the gap function, using the perturbation theory around the reference disks. Starting from disks, we observe the effects of the deformation of the shapes of particles by the n-th-order term of the Fourier series $\sin (n\theta )$. We show that the several physical quantities, such as the number of contacts, gap distribution, and characteristic frequencies of the vibrational density of states, show the power-law behaviors with respect to the linear deviation from the reference disks. The power-law behaviors do not depend on n and are fully consistent with the mean-field theory of the jamming of non-spherical particles. This result suggests that the mean-field theory holds very generally for nearly spherical particles.

Graphic abstract

Appendices

Appendix A: Isostaticity of particles consisting of spherical particles

To keep the generality, we consider N particle system connected by M bonds. For instance, N/2 dimers can be considered as N spherical particles with $M=N/2$ bonds. We consider the harmonic potential:

$$\begin{aligned} V_N = \sum _{i<j}^{1,N}\frac{h_{ij}^2}{2}\Theta (-h_{ij}) + k\sum _{a=1}^M \frac{t_{i_aj_a}^2}{2}, \end{aligned}$$

(A1)

where

$$\begin{aligned}&h_{ij} = \left| \varvec{r}_i-\varvec{r}_j\right| -R_i-R_j,\nonumber \\&t_{i_aj_a} = \left| \varvec{r}_{i_a}-\varvec{r}_{j_a}\right| - l_{i_a j_a}. \end{aligned}$$

(A2)

$\varvec{r}_i=\{x_1^i, \dots , x_d^i\}$ denotes the position, and $R_i$ denotes the diameter of the i-th particle, and $l_{i_aj_a}$ denotes the length of the a-th bond connecting particles $i_a$ and $j_a$.

Here, we show that the system is isostatic at $\varphi _J$ by using the same argument for frictionless spherical particles [11, 47].

The number of degrees of freedom of the system is

$$\begin{aligned} N_f = Nd. \end{aligned}$$

(A3)

At the jamming transition point, we have

$$\begin{aligned}&\left| \varvec{r}_{i_\mu }-\varvec{r}_{j_\mu }\right| = R_{i_\mu }+R_{i_\mu },&\mu = 1,\dots , N_c,\nonumber \\&\left| \varvec{r}_{i_a}-\varvec{r}_{j_a}\right| = l_{i_aj_a},&a = 1,\dots , M. \end{aligned}$$

(A4)

where $N_c$ denotes the number of contacts, $i_\mu$ and $j_\mu$ denote particles of the $\mu$-th contact. One can find $\{\varvec{r}_i\}_{i=1,\dots , N}$ satisfying the above equation if

$$\begin{aligned} Nd \ge N_{\mathrm{const}}, \end{aligned}$$

(A5)

where

$$\begin{aligned} N_{\mathrm{const}} = N_c +M, \end{aligned}$$

(A6)

denotes the number of constraints at the jamming transition point. On the contrary, the force balance requires

$$\begin{aligned} \frac{\partial V_N}{\partial \varvec{r}_i} = \sum _{j\ne i}\Theta (-h_{ij})h_{ij}\varvec{n}_{ij} + k\sum _{a=1}^M t_{ij_a}\varvec{n}_{ij_a} = 0, \end{aligned}$$

(A7)

where $\varvec{n}_{ij}$ denotes the normal vector connecting particles i and j. This can be regarded as Nd linear equations for $\{h_{i_\mu j_\mu }\}_{\mu =1,\dots , N_c}$ and $\{t_{i_aj_a}\}_{a=1,\dots , M}$. One can find a solution if

$$\begin{aligned} N_c+M \ge Nd. \end{aligned}$$

(A8)

From Eqs. (A5) and (A8), we get

$$\begin{aligned} N_{\mathrm{const}} = N_f \leftrightarrow N_c + M = Nd, \end{aligned}$$

(A9)

meaning that the system is isostaticity at the jamming transition point. For N/2 dimers, the total number of contacts is written as $N_c = (N/2)z_J/2$, leading to

$$\begin{aligned} z_J = 4d-2, \end{aligned}$$

(A10)

which is consistent with the numerical results in $d=2$ [29, 30] and $d=3$ [31].

Appendix B: Derivation of Eq. (34)

Fig. 13

Schematic picture of two non-spherical particles. Red solid lines denote particles shape, and blue dashed lines denote reference disks

We write the gap function $h_{ij}$ as

$$\begin{aligned} h_{ij}&= \left| \varvec{u}_i-\varvec{u}_j\right| , \end{aligned}$$

(B1)

where $\varvec{u}_i$ and $\varvec{u}_j$ are points on the surfaces of particles i and j that minimize $h_{ij}$, see Fig. 13. We expand $\varvec{u}_i$ and $\varvec{u}_j$ from those of the reference disks as

$$\begin{aligned}&\varvec{u}_i = \varvec{r}_i + R_i(\varvec{n}_i)\varvec{n}_i = \varvec{u}_i^0 + \delta \varvec{u}_i,\nonumber \\&\varvec{u}_j = \varvec{r}_j + R_j(\varvec{n}_j)\varvec{n}_j = \varvec{u}_j^0 + \delta \varvec{u}_j, \end{aligned}$$

(B2)

where

$$\begin{aligned}&\varvec{u}_i^0 = \varvec{r}_i + R_i^0\varvec{n}_i^0,\ \varvec{u}_j^0 = \varvec{r}_j + R_j^0\varvec{n}_j^0,\nonumber \\&\delta \varvec{u}_i = R_i(\varvec{n}_i)\varvec{n}_i - R_i^0\varvec{n}_{i}^0,\delta \varvec{u}_j = R_j(\varvec{n}_j)\varvec{n}_j - R_j^0\varvec{n}_{j}^0,\nonumber \\&\varvec{n}_{i} = \frac{\varvec{r}_i-\varvec{u}_i}{\left| \varvec{r}_i-\varvec{u}_i\right| },\varvec{n}_{j} = \frac{\varvec{r}_j-\varvec{u}_j}{\left| \varvec{r}_j-\varvec{u}_j\right| },\nonumber \\&\varvec{n}_{i}^0 = \frac{\varvec{r}_i-\varvec{r}_j}{\left| \varvec{r}_i-\varvec{r}_j\right| }, \varvec{n}_{j}^0 = -\varvec{n}_i^0, \end{aligned}$$

(B3)

see Fig. 13. $R_i(\varvec{n}_i)$ denotes the radius of particle i along the direction $\varvec{n}_i$. In particular,

$$\begin{aligned} R_i(\varvec{n}_i^0) = R_i^0\left[ 1 + \Delta F(\theta _i-\theta _{ij})\right] , \end{aligned}$$

(B4)

where $\theta _i$ denotes the direction of particle i, $\theta _{ij}$ denotes the relative angle between particles i and j, see Fig. 3. For $\Delta \ll 1$, we can expand $h_{ij}$ w.r.t $\Delta$ as

$$\begin{aligned} h_{ij}&= \left| \varvec{u}_i-\varvec{u}_j\right| = h_{ij}^0 + \varvec{n}_i^0\cdot \left( \delta \varvec{u}_i - \delta \varvec{u}_j\right) + O(\delta \varvec{u}_i^2,\delta \varvec{u}_j^2)\nonumber \\&= h_{ij}^0 + R_i^0-R_i(\varvec{n}_i) + R_j^0-R_j(\varvec{n}_j) + O(\Delta ^2)\nonumber \\&= h_{ij}^0 - \Delta \left[ R_i^0 F(\theta _i-\theta _{ij}) + R_j^0 F(\theta _j-\theta _{ji})\right] + O(\Delta ^2), \end{aligned}$$

(B5)

where we used $\varvec{n}_i^0\cdot \varvec{n}_i = 1 + O(\Delta ^2)$, and $R_i(\varvec{n}_i) = R_i(\varvec{n}_i^0) + O(\Delta ^2)$. $h_{ij}^0$ denotes the gap function of the reference disks:

$$\begin{aligned} h_{ij}^0 = \left| \varvec{u}_i^0-\varvec{u}_j^0\right| = \left| \varvec{r}_i-\varvec{r}_j\right| -R_i^0-R_j^0. \end{aligned}$$

(B6)

Appendix C: Sorted eigenvalues

Fig. 14

Sorted eigenvalues of the Hessian matrix $H_{X_iY_j}$. We used the same parameters as in Fig. 11. $N_\lambda$ denotes the number of the eigenvalues. The separate bands are visible at around $i/N_\lambda \sim 0.3$

In Fig. 14, we show the (sorted) eigenvalues $\lambda _i$ of the Hessian matrix Eq. (45). We used the same parameters n and $\Delta$ of those of Fig. 11. Here, we only show the data for $\lambda _i>10^{-12}$, which is large enough to remove the zero modes. As in the case of Fig. 11, the separate band arises at around $i/N_\lambda \sim 0.3$.