Research Interests

Research Interest Next

I. Mean-Field Thoery of Glass Transitions

We consider the following two kinds of systems;
(A) Suspensions of colloidal particles, including neutral, magnetic, and charged colloids),
(B) Molecular systems with any types of potentials.

Let $\bm{X}_i(t)$ denote the position vector of the $i$th particle at time $t$. Then, the mean-square displacement $M_2(t)$ is given by

$$M_2(t)=\frac{1}{N}\sum_{i=1}^N<[\bm{X}_i(t)-\bm{X}_i(0)]^2>,$$ (1)

where $N$ denotes the total number of particles. As discussed in Ref. [1], it obeys the following mean-field equation:
(A) Suspensions of colloids

$$\frac{d}{dt}M_2(t)=2dD_S^L(p)+2d\left[D_S^S(p)-D_S^L(p)\right]e^{-M_2(t)/\ell(p)^2},$$ (2)

(B) Molecular systems

$$\frac{d}{dt}M_2(t)=2dD_S^L(p)+2d\left[\frac{v_0^2}{d}t-D_S^L(p)\right]e^{-M_2(t)/\ell(p)^2},$$ (3)

where $\ell(p)$ denotes the free length, during which the particles do not interact, $p$ the control parameter, $v_0(=\surd\overline{dk_BT/m})$ the average velocity, $D_S^L(p)$ the long-time self-diffusion coefficient, and $D_S^S(p)$ the short-time self-diffusion coefficient. Here $T$ is a temperature of the system and $m$ the mass of a particle. Equations (2) and (3) are easily solved to give

$$M_2(t)=\ell^2\ln\left[1+\frac{D_S^S}{D_S^L}\left\{e^{2dD_S^Lt/\ell^2}-1\right\}\right],$$ (4)

$$M_2(t)=\ell^2\ln\left[1+2\left(\frac{\ell v_0}{2dD_S^L}\right)\left\{e^{2dD_S^Lt/\ell^2}-1-2dD_S^Lt/\ell^2\right\}\right],$$ (5)

respectively. In Figs. 1, 2 , and 3, the mean-field results are compared with the experimental results for the suspension by van Megen et al [2], with the simulation results for the hard-sphere fluid by Tokuyama and Terada [3,4], and with the simulation results by Gallo, et al [5], respectively. In former two cases, the control parameter is given by the particle volume fraction $\phi$, while in latter case, the temperature $T$ is a control parameter. The fitting is done by adjusting the free length $\ell$ and the long-time self-diffusion coefficient $D_S^L$ at each volume fraction. In Figs. 4 and 5 the fitting values of $\ell(\phi)$ and the free volume $V_f(=\ell^d)$ are plotted versus $\phi$, respectively. For comparison, the free length $\ell_c$ and the free volume $V_f^c(=\ell_c^d)$ of crystal are also plotted, where

$$\ell_c=\frac{1}{\surd\overline{2}}\left(\frac{2\pi}{3\phi}\right)^{1/3}-1.$$ (6)

In Fig. 6 the fitting values of the long-time self-diffusion coefficient$D_S^L$ are plotted versus the volume fraction $\phi$.

Fig. 1: Comparison between the experimental results and the mean-field results. The red solid lines indicate the experimental results and the blue solid lines the mean-field results.

Fig. 2: Comparison between the simulation results for the hard-sphere fluid and the mean-field results. The red solid lines indicate the simulation results and the blue solid lines the mean-field results.

Fig. 3: Comparison between the simulation results for the Lennard-Jones liquids and the mean-field results. The red solid lines indicate the simulation results and the blue solid lines the mean-field results.

Fig. 4: The free length versus the volume fraction. The red circles indicate the experimental results, the blue circles the simulation results, and the green line the free length of crystal.

Fig. 5: The free volume versus the volume fraction. The details are the same as in Fig. 4.

Fig. 6: The long-time self-diffusion coefficient versus the volume fraction. The red circles indicate the experimental results and the blue circles the simulation results. The solid lines indicate the non-singular functions of $\phi$ predicted by Tokuyama [1,6]. The difference between coefficients is mainly due to the existence of the long-time hydrodynamic interactions in the experiment [1,3].

References

1

M. Tokuyama, Physica A, 364, 23-62 (2006).

2

W. van Megen, T. C. Mortensen, S. R. Williams, J. Müller, Phys. Rev. E 58, 6073 (1998).

3

M. Tokuyama, H. Yamazaki, and Y. Terada, Phys. Rev. E 67, 062403 (2003).

4

M. Tokuyama and Y. Terada, J. Phys. Chem. B 109, 21357 (2005); to be published in AIP Conference Series (July, 2006); to be submitted to J. Phys. Condensed Matters (January, 2006).

5

P. Gallo, R. Pellarin, and M. Rovere, Phys. Rev. E 67, 041202 (2003).

6

M. Tokuyama, Physica A 289, 57 (2001).

Home

Research Interest

Next