Self-assembly processes of superparamagnetic colloids in a quasi-two-dimensional system

Superparamagnetic colloids gather depending on the magnitude of the magnetic field applied, forming chains and ribbons in a quasi-two-dimensional chamber. The results presented in this work are in good agreement with recent experimental multistable data for the mean length of the aggregates in thermodynamic equilibrium.

Figure 1

Mean length of saturated chains and ribbons at thermodynamic equilibrium. Experimental data obtained from Refs. [12, 13] [(blue) $\left[\right]$]. Solution of Eq. (10) $\overline{s}=\sqrt{{\varphi }_0{e}^{\beta {\epsilon }_m}}$ (black thick line) [14]. Upper limit provided by central formula in inequality (11) (upper dotted line). Lower bound given by Eq. (12) (lower dotted curve). Equation (13) for ${\overline{s}}_{\infty }=35$, 50, 70, 100, 140 (gray dashed curves). Complete first order solution (27) (gray thick curve) and previous approximations for ${\varphi }_0=4.41$. Solutions ${\overline{s}}_{a1}$ (33) (dot-dashed line), (34) for ${\overline{s}}_{\infty }=100$, 140 (black dashed curves), and (35) for ${\overline{s}}_{\infty }=50,100$ (black thin curves) evaluated at ${\varphi }_0=0.81$.

Figure 2

Upper limit provided by central formula in inequality (11) (upper dotted line). Lower bound given by Eq. (12) (lower dotted curve). Solution of Eq. (10) (black thick line) using (8) [14]. Equation (13) for ${\overline{s}}_{\infty }=100$ (gray dashed curve).

Figure 3

Symmetric aggregates of length $s=4,5$ and width $d=1,2,3,4$ oriented in the direction of the magnetic field. The number of first order bonds (short lines) is given by $n_b^{\left(1\right)}=3(n-1)-p=d(s-1)+(d-1)(2s-1)$; the amount of second order bonds (long lines) is $n_b^{\left(2\right)}=(d-1)(2s-3)+(d-2)s$. Perimeter grains $p$ are filled (blue color). Negative potential interactions are presented with solid (blue) lines and positive potential interactions with dashed lines.

Figure 4

Asymmetric ribbons labeled in nondimensional coordinates $({\rho }_{i,x},{\rho }_{i,y})$ for $d=2,3,4$ and $s=4$. The rest $n\text{mod}d$ particles are filled (blue). The particle $n_0=d^2$ (dashed line circles) indicates the smallest ribbon aligned parallel to the direction of the magnetic field, considered in the fit of the numerical results (29) employing $u_{d,n}=2(a_d-b_{d}n)/n$.

Figure 5

Normalized magnetic energy per particle $u(d,n)$ of one chain or ribbon computed employing Eq. (29) up to $n=700$ at $d=1$ (circles), $d=2$ [(blue) $\left[\right]$], $d=3$ [(red) $\#$], and $d=4$ (gray circles). Fit $u_{d,n}=2(a_d-b_{d}n)/n$ for $d=1$ (gray line), $d=2$ (dashed curve), $d=3$ (dot-dashed line), and $d=4$ (dotted line). Inset: $u(d,n)$ computed up to $n=500$ particles, the most stable aggregate changes at about $n_d=30$, 113, 263 for $d=2,3,4$ respectively.

Figure 6

Mean length of saturated chains and ribbons at thermodynamic equilibrium. Solutions $\overline{s}=\sqrt{{\varphi }_0{e}^{\beta {\epsilon }_m}}$ (black thick line) [12, 14], (27) (gray thick curve), and ${\overline{s}}_{a1}$ (33) [dot-dashed (red) line] evaluated at ${\varphi }_0=0.81$. Absolute value of numerical solutions obtained using (31) and (32) up to $d_{\max }=2$ [(blue) points]. Inset: Absolute value of numerical solutions up to $d_{\max }=3$ ((blue) points).

Figure 7

Mean length of saturated chains and ribbons at thermodynamic equilibrium. Absolute value of numerical solutions obtained using (31) and (32) up to $d_{\max }=4$ for ${\varphi }_0=0.81$ [(green) $\#$]. Absolute value of numerical solutions for ${\varphi }_0=0.09$ (circles). Inset: Numerical solutions diverge as $B$ increases. The magnitude of $|\overline{s}|$ grows with ${\varphi }_0$ [24].

Figure 8

Mean length of saturated chains and ribbons at thermodynamic equilibrium. Experimental data obtained from Refs. [12, 13] [(blue) $\left[\right]$]. Absolute value of selected solutions ${\overline{s}}_4$ to ${\overline{s}}_{12}$ (circles) and completely real numerical data from solutions ${\overline{s}}_4$ to ${\overline{s}}_{18}$ [(green) $\#$] using (31) and (37) up to $d_{\max }=4$ for ${\varphi }_0=0.002$ and $C_1=0.2\times {10}^{-2}$ $\left[G^{-2}\right]$.