Tangential-force model for interactions between bonded colloidal particles

Recently, Pantina and Furst [Phys. Rev. Lett. 94, 138301 (2005)] experimentally demonstrated the existence of tangential forces between bonded colloidal particles and the capability of these bonds to supporting bending moments. We introduce a model to be used in computer simulations that describes these tangential interactions. We show how the model parameters can be determined from experimental data. Simulations using the model are in agreement to the measurement by Pantina and Furst. Application of the model to an aggregate with fractal structure leads to more realistic behavior than using classical approaches only.

Figure 1

Interaction potential between two equal spherical particles of radius $a=0.735\mu \mathrm{m}$. The potential comprises attractive van der Waals interactions with a Hamaker constant of $A=0.062\mathrm{eV}$, repulsive electrostatic interactions in a $150\mathrm{mM}$ $\mathrm{Mg}{\mathrm{Cl}}_2$ aqueous solution, a surface potential of ${\varphi }_0=40\mathrm{mV}$, and the Born repulsion where $N$ is assumed to be $N={10}^{-23}$.

Figure 2

Example for the bending torque, the lower particle is fixed and on the upper particle acts as a tangential force (left-hand side). If the bond between the particles is able to support a bending moment, a stationary angle $\varphi <\pi ∕2$ between the original contact line and the stationary contact line should be reached.

Figure 3

Sketch of tangential-force model. (a) When two particles get in contact, two thought rods are initialized. The rods are rigidly connected to the center of one particle and reach the other particles center. (b) According to the tangential movement of the particles, the springs ${\mathbf{\xi }}_{ij}$ and ${\mathbf{\xi }}_{ji}$ will be elongated. The arrows denote the direction of the vectors ${\mathbf{\xi }}_{ij}$ and ${\mathbf{\xi }}_{ji}$.

Figure 4

Simulated shape of a bended 11-particle aggregate for different material parameters. The used tangential stiffness are $k_t=0.785\mathrm{mN}∕\mathrm{m}$ (circles), $k_t=1.1\mathrm{mN}∕\mathrm{m}$ (crosses), $k_t=1.7\mathrm{mN}∕\mathrm{m}$ (squares), and $k_t=3.4\mathrm{mN}∕\mathrm{m}$ (diamonds). The symbols are simulation results and the lines are obtained from (1).

Figure 5

Typical deflection-force curves for a 11-particle (upper line) and a 23-particle aggregate (lower curve). The parameters for the tangential-force model are $k=0.69\mathrm{mN}∕\mathrm{m}$ and ${\xi }_{\max }=30.48\mathrm{nm}$ obtained from measured data in [21] for a 150 $\mathrm{Mg}{\mathrm{Cl}}_2$ solution. The dashed line shows the experimentally obtained linear part of the deflection-force curve taken from a data plot (Fig. 2) presented by [21].

Figure 6

(Color online) Compaction of an aggregate using central forces only. (a) The start configuration, generated by diffusion-limited aggregation. (b) The configuration after a simulated time period of $25\mathrm{s}$, the structure is significantly more compact than the starting configuration.

Figure 7

Time evolution of the average number of neighbors. The solid line shows simulation results where DLVO forces only were considered. The dashed line displays data obtained from simulations where the tangential-force model was applied.

Figure 8

Sketch for the calculation of the bending energy. The initial configuration is shown by solid lines. After the deformation process the three particles have new positions and orientations, shown by the dashed lines. The particles $l$ and $j$ are reoriented relative to particle $i$ by the angles $\delta {\varphi }_l$ and $\delta {\varphi }_i$, and particle $i$ was rotated around its center by the angle $\delta {\phi }_i$. The springs ${\xi }_{ij}$ and ${\xi }_{il}$ are deformed due to the particle motion.

Figure 9

Sketch for the calculation of the equilibrium shape.