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IV. Slow dynamics of magnetic colloidal chains confined in thin film

We investigate the slow dynamics in the suspension of the magnetic colloidal chains confined in a thin film [1].
We consider the model system confined in a thin film with thickness $ L_z$ which contains $ N_{xy}$ magnetic colloidal chains dispersed in an equilibrium solvent with a viscosity $ \eta$ at temperature $ T$. The external magnetic field $ \bm{H}$ is applied perpendicular to the film.
Fig. 1: A Snapshot of magnetic colloidal chains confined in thin film.
\includegraphics[height=0.45\linewidth]{chain-poly.ps}

The position of the center of mass for $ \alpha$-th magnetic colloidal chain $ \bm{X}_\alpha(t) $ is described by the Langevin-like equation on the time scale $ t_c(=a^2/D_c)$

$\displaystyle \frac{d}{dt}{\bm{X}}_\alpha(t)$ $\displaystyle =$ $\displaystyle \frac{1}{ \underset{i=1}{\overset{N_z^\alpha}{\sum}} M_{i}^\alpha... ...}^{\alpha\beta}, t) M_{i}^\alpha}{\gamma_i ^\alpha} + {\bm{f}}_G^\alpha(t) $ (1)

where $ t_c$ is the relaxation time for the colloidal chain to diffuse a distance of average radius $ a$ for the colloidal chains and $ D_c$ an average diffusion coefficient of a single chain. $ {\bm{f}}_G^\alpha(t)$ is a Gaussian, Markov random velocity with zero mean and $ \bm{F}$ a dipole force between particle $ i$ in the $ \alpha$-th chain and particle $ j$ in the $ \beta$-th chain. Here the $ \alpha$-th colloidal chain consists of $ N_z^\alpha$ magnetic colloidal particles and the colloidal particle $ i$ in $ \alpha$-th colloidal chain has the susceptibility $ \chi_i^\alpha$, mass $ M_i^\alpha$, and radius $ a_i^\alpha$. Let $ N_z$ denote the average number of the colloidal particles which form one colloidal chain. The area fraction of the colloidal chains $ \sigma$ is given by $ \sigma = \pi \sum_{\alpha=1}^{N_{xy}}a_1^\alpha ^2/S $ where the total area of the film is $ S$.

Figure 2 shows a two-dimensional projection of the magnetic colloidal chains on xy plane for (a) monodisperse system, (b) binary systems, and (c) polydisperse system. Here the dimensionless parameter $ \Gamma (=\frac{4}{9} \frac{\pi a^3 \mu_0 \chi_0^2 H^2 }{ k_BT} \sigma ^{3/2})$ is the indicator of the intensity of the dipole potential energy [2].

Fig. 2: A projection of magnetic colloidal chains on xy plane for binary system, polydisperse system, and monodisperse system.

Figure 3 shows the mean square displacement of chains given by $ M_2(t)=< (\bm{X}_\alpha(t)-\bm{X}_\alpha(t=0) )^2>$ for (a) the binary colloidal suspensions with number ratio of the small colloidal chains to all chains $ \omega=0.1$, where we set the parameters of the particles the same as those in ref.[2] and the small (or big) colloidal chain consists of all identical small (or big) particles, and for (b) the 15 % polydisperse colloidal suspensions. On both suspensions of colloidal chains confined in the film with different thickness, the crystallization does not occur as the intensity of the external field is increases, where no long-range order exists at the strong external field. The only difference in those systems with different thickness is that the average long-time diffusion coefficient of all chains confined in the thin film is smaller than that in the thick film at a constant external field. The reason is that many-body interactions between the long chains confined in the thick film is much stronger than those in the thin film. We also find that the dynamic behaviour of the systems with the same average long-time diffusion coefficient is described by one master curve, even though those confined in the film with different thickness contain both of binary colloidal chains with any number ratio and polydisperse systems. It is one example for similarities in diversely different glass-forming systems [3].

Fig 3: A log-log plot of the mean-square displacement vs time. (a) monolayer binary colloidal particles with $ \omega =1$ for $ \Gamma =10.00, 55.25, 110.5, 221.0$, and $ 331.6$ and for (b) $ N_z=5$ 15 % polydisperse colloidal chains for $ \Gamma =5.196, 10.39, 51.96,$ and $ 103.9$ from top to bottom at $ \sigma =0.03$. Dashed lines indicates the simulation results on liquid state and solid lines supercooled liquid state.
(a) (b)
\includegraphics[height=0.485\linewidth]{figa-1-binary-M2.eps} \includegraphics[height=0.485\linewidth]{figa-1-poly-M2.eps}



References
1 Y. Terada, thesis (Tohoku University) (2007).
2 H. König, R. Hund, K. Zahn, and G. Maret, Euro. Phys. J. E 18, 287 (2005).
3 M. Tokuyama, Physica A (2007) "Similarities in diversely different glass-forming systems" Available online 10 January 2007.

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