Directed assembly of active colloidal molecules

For the construction of the cilia- and flagella-like structures described in [19], we considered a simple model of active particles, which do not self-propel on their own, but rather activity emerges as a result of interactions [27]. In presence of concentration inhomogeneities, colloids experience a phoretic drift which is well described by a velocity proportional to the concentration gradient of a relevant solute, multiplied by a mobility coefficient μ, $\vec{V}=-\mu {\rm{\nabla }}c$ [28]. This concentration field can be generated by external sources and sinks ${c}^{\mathrm{ext}}(\vec{r})$, but also it can be produced by catalytic reactions on the particles surfaces themselves. Active colloidal particles, for example, gold or platinum particles in a water peroxide solution, generate or consume solutes at their surfaces, which later diffuse on the fluid. Janus particles self-propel by this mechanism [29, 30]. Here, we consider isotropic colloids, each one made of a pure material (e.g. gold or platinum). For micron sized particles, with speeds of few microns per second, the solute Péclet number is small; hence, we use the approximation where the solute concentration profiles are completely enslaved to the instantaneous positions of the colloids ${\vec{r}}_{i}$ and the production rates at their surfaces α_i. The case of finite Péclet number gives rise to interesting fenomena as it has been reported in [31]. In the far field approximation, the interaction between two colloids in three dimensions decays as the distance squared and the resulting diffusiophoretic velocity of the colloid i interacting with colloid k is ${\vec{V}}_{i}={U}_{0}{\mu }_{i}{\alpha }_{k}{\hat{r}}_{{ki}}/{r}_{{ki}}^{2}$, where ${\vec{r}}_{{ki}}={\vec{r}}_{i}-{\vec{r}}_{k}={r}_{{ki}}{\hat{r}}_{{ki}}$ is the vector that joins the centers of the particles and U₀ depends on their radii and the diffusion coefficient of solutes [27]. Notably, the interaction between two dissimilar particles is not symmetric, violating the action-reaction principle. This is possible by the presence of the supporting fluid that takes away or provides momentum, and it is the source of activity in this model. Besides this interaction, colloidal particles display steric repulsion, which we model as a soft-sphere potential. The equations of motion for overdamped particles in presence of Brownian noise read

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{\vec{r}}_{i}}{{\rm{d}}{t}}={\vec{V}}_{i}+\eta {\vec{\xi }}_{i}(t),\end{eqnarray} \tag{ 1 }$$

with

$$\begin{eqnarray}&&{\vec{V}}_{i}={U}_{0}\displaystyle \sum _{k\ne i}{\alpha }_{k}{\mu }_{i}\displaystyle \frac{{\hat{r}}_{{ki}}}{{r}_{{ki}}^{2}}+{U}_{1}\displaystyle \sum _{k\ne i}\displaystyle \frac{{\hat{r}}_{{ki}}}{{r}_{{ki}}^{12}}-{\mu }_{i}{\rm{\nabla }}{c}^{\mathrm{ext}},\end{eqnarray} \tag{ 2 }$$

where U₁ measures the intensity of the repulsive force and η is the noise intensity, which gives rise to a bare diffusion coefficient for monomers, D = η²/2. Here, ${\vec{\xi }}_{i}(t)$ is a vectorial white noise of zero mean and correlation $\langle {\xi }_{i,a}(t){\xi }_{k,b}(t^{\prime} )\rangle ={\delta }_{{ik}}{\delta }_{{ab}}\delta (t-t^{\prime} )$, with i, k labeling particles and a, b indicating the Cartesian coordinates. Although the pair interaction has been derived in three spatial dimensions, as colloidal particles normally sediment, their motion will be restricted to two dimensions.

For a binary mixture (types A and B) with the signs of the phoretic charges chosen such that (α_A, μ_A) = (+, +) and (α_B, μ_B) = (−, −), equal particles repel and dissimilar particles attract. At low concentration of colloidal particles, this results in the formation of clusters or colloidal molecules, where A and B particles alternate. The smallest of these are dimers AB and trimers AB₂. Depending on the values of the charges, colloidal molecules can be stable in different isomer configurations. For example, the trimer can adopt a linear configuration, but also, for a wide range of parameters, it adopts a polar triangular shape. The dimer is also polar and both present self-propulsion as a result of the violation of the action-reaction principle [27].

It is possible to absorb the dimensions of α and μ into U₀, allowing us to chose α_B = −1 and μ_B = −1, leaving α_A and μ_A (both positive) as free dimensionless variables. Also, time and space units can be chosen such that U₀ = U₁ = 1. Alternatively, keeping dimensional forms for U₀ and U₁, the length, time, and velocity units are ${l}_{0}={({U}_{1}/{U}_{0})}^{1/10}$, ${t}_{0}={U}_{1}^{2/5}/{U}_{0}^{7/5}$, and ${v}_{0}={l}_{0}/{t}_{0}={U}_{0}^{13/10}/{U}_{1}^{3/10}$, respectively. The minimal distance between particles that attract each other is of order l₀, therefore fixing the hard-core diameter. For comparison, in the experiments described in [25], l₀ ≈ 10 μm and v₀ ≈ 1 μm s⁻¹, resulting in t₀ ≈ 1 s. Similar molecules have been found with different choices of the mobilities' signs [25, 26, 32] but we are not aware that the oscillating chain can be formed like that, so we restrict ourselves to the present case.

Strong inhomogeneities on the field can break colloidal molecules by tidal effects (and will be studied in detail in section 4), but at moderate values, colloidal molecules will behave as solid bodies and the field will move and rotate them [22]. For isotropic clusters composed of N particles, the external field produces a net velocity. It is proportional to the concentration gradient evaluated at the geometric center of the cluster, $\vec{R}={\sum }_{i}{\vec{r}}_{i}/N$, and the cluster's average mobility, $\overline{\mu }={\sum }_{i}{\mu }_{i}/N$, plus corrections that depend on the second derivatives of c^ext. For anisotropic clusters, besides this velocity, there is an induced angular velocity $\vec{\omega }$, which is computed as follows. Defining the relative coordinates ${\vec{x}}_{i}\equiv {\vec{r}}_{i}-\vec{R}$, it is direct that $\dot{{\vec{x}}_{i}}=-({\mu }_{i}-\overline{\mu }){\rm{\nabla }}{c}^{\mathrm{ext}}$ where the gradient is evaluated at $\vec{R}$ by neglecting contributions from higher derivatives. Assuming that the cluster does not deform or break thanks to the balance of the phoretic and repulsive forces, the solid body rotation implies also, $\dot{{\vec{x}}_{i}}=\vec{\omega }\times {\vec{x}}_{i}$. Equating both expressions and using that in two dimensions ${\vec{x}}_{i}$ and $\vec{\omega }$ are perpendicualr, gives $\vec{\omega }=-\vec{p}\times {\rm{\nabla }}{c}^{\mathrm{ext}}$, where $\vec{p}={\sum }_{i}({\mu }_{i}-\overline{\mu }){\vec{x}}_{i}/{\sum }_{i}| {\vec{x}}_{i}{| }^{2}$ is the dipolar moment of the cluster. How fast these molecules align and what trajectory they follow depend on the geometry of the molecule; thus, for example, dimers turn much faster than trimers in an external field (see figure 1). These formulas are for illustration purposes only, we actually integrate the particle-particle interaction in an external field for field strengths small enough as to not break any of the formed molecules.

Zoom In Zoom Out Reset image size

For the efficient production of dimers and trimers, we take an external concentration field with a geometry that focuses particles in a small reaction zone. This field is assumed to be produced by sources and sinks far from the reaction zone. For this, it is chosen to be a solution of the stationary diffusion equation, built using the conformal mapping technique. There are two impenetrable walls that leave a small opening of width 2b₀, where monomers and colloidal molecules can interact (see figure 2). We take

$$\begin{eqnarray}&&{c}^{\mathrm{ext}}(x,y)={c}_{0}\mathrm{Im}\left(\sqrt{{b}_{0}+x+{\rm{i}}{y}}-\sqrt{{b}_{0}-x-{\rm{i}}{y}}\right),\end{eqnarray} \tag{ 3 }$$

where Im is the imaginary part and c₀ measures the field intensity. To avoid colloids to reach and cross the walls, an additional repulsive potential is added

$$\begin{eqnarray}&&{P}_{\mathrm{wall}}={P}_{0}\displaystyle \frac{{x}^{2}-{b}_{0}^{2}}{{x}^{2}{y}^{6}+{r}_{0}},\end{eqnarray} \tag{ 4 }$$

where P₀ = r₀ = 1, such that the induced velocity on each colloid is ${\vec{V}}_{\mathrm{wall}}=-{\rm{\nabla }}{P}_{\mathrm{wall}}$ (equal for all particles).

Zoom In Zoom Out Reset image size

If A and B monomers are inserted at the rectangular regions in figure 2, located at opposite sides of the reaction zone, they will move along the field lines and meet near the origin where they can interact and assembly into colloidal molecules. Since we aim for particles to arrive at the origin at roughly the same time, we need to put particles at different locations, for they have different mobilities. Each particle is injected with a uniform probability in a rectangle of size 2 × 6 in the x and y direction, respectively. This mimics particles that are let to sediment from the third dimension into these regions in microfluidic devices. The insert location is such that the particles tend to meet at the center of the system. The distance between the two insertion points permits the particles to first form a dimer, then change their direction of movement, and after that collide with another monomer to form a trimer. See supplementary video 1 available online at stacks.iop.org/NJP/21/033041/mmedia for an example.

The results presented here are in the diluted-injection regime, that is, there are only a few monomers in the volume at any time, and the times between insertions are comparable to the time it takes a monomer to travel the system due to the effect of the external field only. In this way, we ensure that the majority of reactions in the system are monomer to monomer, and monomer to dimer, and colloidal molecules are always isolated so no larger reaction occurs.

Since we aim to simulate the continuous production of colloidal molecules, the total number of particles in our simulation is not conserved: as we inject particles their number increases, and for the sake of simplicity, we remove them when they are far enough from the production zone. The insertion of particles happens at fixed time intervals Δt_i, where i = A, B indicates the species of the particles. Particle removal occurs when the first particle of a cluster crosses the circular boundary of radius R = 50. Note that dimers and trimers leave the system by self-propulsion whereas the monomers move passively following the external field lines as can be seen in figure 1.

Thanks to the external field, the monomers do not stick around the injection area and collapse into large amorphous clusters of particles. With the presence of the external field the unused monomers will leave the reaction area, facilitating the organized formation of dimers or trimers alone.

In the regime we are studying, the most important control parameters are the fluxes of incoming particles, J_i = 1/Δt_i, which determine what molecules are formed. Figure 3 presents the fraction of monomers, dimers and trimers that exit the system as a function of the flux ratio J_B/J_A. Simulations were performed with η = 0. Note that for the zero noise case, corresponding to a vanishing temperature, if the system were not driven but let to relax freely, it would have collapsed to a single large cluster [27]. Here, the driving imposed by the external field allows us to steadily produce small colloidal molecules.

Zoom In Zoom Out Reset image size

When the fluxes of A and B particles are the same, the system forms mostly dimers. By increasing the flux of B particles, the chances to form trimers increase. One would expect that a flux of B particles twice larger will create mostly trimers, but this is actually not the case. For J_B = 2J_A, many B colloids escape as monomers, without reacting to form colloidal molecules. The maximum production of trimers occurs for J_B ≈ 1.7J_A, in which case, 60% of the collected clusters are trimers.

The angular distribution of the produced colloidal molecules is shown in figure 4. Notably, dimers and timers are well focused and separated from monomers. A further separation between dimers and trimers could be arranged adding a perpendicular component to the external field. To have an idea of how a perpendicular field separates dimers of trimers is enough to look at figure 1 and how dimmers and trimers that are very close at one point during their trajectory (lower part of the figure) are separated thanks to the external field. By separating these small molecules one can use them to produce larger functional structures, as shown in the next sections.

Zoom In Zoom Out Reset image size

The results presented in this section are qualitatively robust for a wide range of parameters. We have changed the insert location, field strength, opening between the walls and chemistry of the particles.

The values b = 2 and c₀ = 0.525 used for the simulation shown in figures 3 and 4 were chosen after an exploration of different possibilities to produce dimers and trimers with high yield and angular focalization for the specified values of the phoretic charges. Changing b₀ varies the production fractions, but no systematic tendency was detected. However, increasing b₀ widens the angular distribution, losing the focalization shown in figure 4. Increasing c₀ reduces the production of trimers because the monomers are driven too fast and escape the reaction zone before meeting another monomer, while decreasing its value leads to an accumulation of monomers in the reaction zone, resulting in the formation of large clusters. This problem is solved if the fluxes are reduced in proportion.

Finally, other values of the phoretic changes α_A and μ_A in the range of parameters where dimers are self-propelled and trimers are stable (see [27]), have the same qualitatively results. The values of the optimal field strengths, opening size and flux ratios change accordingly. The addition of moderate noise does not change this picture qualitatively since monomers are carried by the external field towards the reaction area.

In [32], it was shown that Janus particles can be guided in chemical landscapes. Here, we show that the same is possible in self-assembled colloids, with the precaution that the chemical gradient should not break the colloidal molecules apart. Consider an AB dimer moving in a external concentration field c^ext(x, y) = λy²/2. We assume for the moment that λ is small enough not to break the dimer and, a posteriori, we determine the maximum allowed value. The velocity of each colloid, which results from the mutual interaction and with the external field is

$$\begin{eqnarray}&&{\vec{V}}_{A}={V}_{0}{\alpha }_{B}{\mu }_{A}\hat{n}+{V}_{1}\hat{n}-{\mu }_{A}\lambda (y-{r}_{\mathrm{eq}}\sin \theta /2)\hat{y},\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{\vec{V}}_{B}=-{V}_{0}{\alpha }_{A}{\mu }_{B}\hat{n}-{V}_{1}\hat{n}-{\mu }_{B}\lambda (y+{r}_{\mathrm{eq}}\sin \theta /2)\hat{y},\end{eqnarray} \tag{ 6 }$$

where $\hat{n}=\cos \theta \hat{x}+\sin \theta \hat{y}$ is the unit vector pointing from B to A, y is the coordinate of the center of mass, ${V}_{0}={U}_{0}/{r}_{\mathrm{eq}}^{2}$ and ${V}_{1}={U}_{1}/{r}_{\mathrm{eq}}^{12}$, with r_eq the equilibrium distance of the two colloids (of order 1). We assume as in [19, 27] that the radial distance is a hard degree of freedom, which does does not vary even in presence of an external field. Imposing that the relative velocity $\dot{\vec{\rho }}={\vec{V}}_{A}-{\vec{V}}_{B}$ vanishes in the $\hat{n}$ direction, gives ${V}_{1}=({\mu }_{A}-{\mu }_{B})\lambda y\sin \theta -{V}_{0}({\alpha }_{A}{\mu }_{B}+{\alpha }_{B}{\mu }_{A})$, where for simplicity we assumed that r_eq ≪ y. Then, analyzing the velocity of the center of mass and the angular component of the relative velocity gives the coupled equations

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}\tilde{y}}{{\rm{d}}\tilde{t}}=({\alpha }_{A}-{\mu }_{A})\sin \theta -\tilde{\lambda }\tilde{y},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}\theta }{{\rm{d}}\tilde{t}}=-\tilde{y}\cos \theta ,\end{eqnarray} \tag{ 8 }$$

where $\tilde{t}=\sqrt{\lambda {V}_{0}({\mu }_{A}+1)/(2{r}_{\mathrm{eq}})}t$ and $\tilde{y}=\sqrt{2\lambda ({\mu }_{A}+1)/({V}_{0}{r}_{\mathrm{eq}})}y$ are rescaled variables and $\tilde{\lambda }=\sqrt{\lambda {r}_{\mathrm{eq}}/[2{V}_{0}({\mu }_{A}+1)]}({\mu }_{A}-1)$ is a control parameter. For α_A − μ_A > 0 and μ_A > 1 ($\tilde{\lambda }\gt 0$), the configuration with the dimer moving along the valley of the concentration field (y = and θ = 0, π) is a stable equilibrium, which is achieved via damped oscillations, as shown in the phase portrait of figure 5. When α_A − μ_A > 0 and μ_A < 1 this configuration is unstable and the dimer escapes from the valley. When α_A − μ_A < 0 the referred configuration becomes a hyperbolic point and the resultant dynamics is not interesting for focusing dimers. Besides the stability, it must be imposed that the tidal forces do not break the dimer. This is guaranteed by requiring that V₁ > 0, which gives $y\sin \theta /{y}_{0}+1\gt 0$, where y₀ = V₀(α_A + μ_A)/[λ(μ_A + 1)]. Figure 5 displays the excluded regions where the tidal forces break the dimer, which are far from the equilibrium configuration. In summary, a concentration field can direct dimers to move along the valley for a wide range of parameters, allowing to use ad hoc potentials to guide colloidal molecules.

Zoom In Zoom Out Reset image size

The analysis was performed for a straight valley, but it is possible to use other geometries to guide different colloidal molecules too. Figure 6 shows the trajectory of a trimer in a circular potential c^ext = λ(y² − R²)², which presents oscillations before reaching the final circular trajectory. Due to the symmetry of the potential the motion could be either clockwise or counter-clockwise, depending on the initial condition. This circular trap also works for dimers in a qualitatively similar way.

Zoom In Zoom Out Reset image size

We have also implemented collisions with walls as a way of guiding the molecules. This has been previously demonstrated for Janus particles [21] but since our molecules are self-assembled and there is no hard forces keeping them together or resisting deformation, it remains to be verified that they will remain a unit when hitting a wall. We implement the wall as a linear restitutive force pointing normal to the surface The hydrodynamic interaction with walls, which despite being long range is far smaller than the elastic force [21], was not included for simplicity. However, our integration method is robust enough to add different long range fields, be it hydrodynamic or phoretic if one would like to add it. Dimers remain stable on collisions and can be guided by straight walls. Eventually, noise can detach dimers from the walls (figure 7-left). For sinusoidal walls the dimer is guided in the convex section and departs when the wall changes curvatures (figure 7-center). Trimers on the other hand, 'stick' to the wall for a moment and then bounce back (figure 7-right).

Zoom In Zoom Out Reset image size