Radial Ratchet

The flexibility of holographic optical thermal ratchet implementations and the success of our initial studies of one-dimensional variants both invite consideration of thermal ratchet operation in higher dimensions. This is an area that has not received much attention, perhaps because of the comparative difficulty of implementing multidimensional ratchets with other techniques. As an initial step in this direction, we introduce a ratchet protocol in which manifolds of traps are organized into evenly spaced concentric rings whose radii advance through a three-state cycle analogous to that in Eq. (13). The probability distribution $p(\vec{r}, t)$ for a Brownian particle to be found within $d\vec{r}$ of $\vec{r}$ at time $t$ under external force $\vec{F}(\vec{r},t) = - \nabla V(\vec{r},t)$ satisfies

$$\frac{\partial p(\vec{r},t)}{\partial t} = D \, \left[\nabla^2 p(... ...ta \, \nabla \cdot \left\{ p(\vec{r},t) \, \vec{F}(\vec{r},t) \right\} \right].$$ (14)

If the force depends only on the radial coordinate as $\vec{F}(\vec{r},t) = -\partial_r V(r, t) \, \hat{\boldsymbol{r}}$, Eq. (14) reduces to

$$\frac{\partial p(r,t)}{\partial t} = D \, \left[ \frac{1}{r} \fra... ...}{r}\frac{\partial}{\partial r} \left\{ r V^\prime(r,t) p(r,t) \right\} \right]$$ (15)

The probability $\rho(r,t)$ for a particle to be found between $r$ and $r+dr$ at time $t$ is given by $\rho(r,t) = 2 \pi r \, p(r,t)$. Therefore, the Fokker-Planck equation can be rewritten in terms of $\rho(r,t)$ as

$$\frac{\partial \rho(r,t)}{\partial t} = D\left[ \frac{\partial^2 ... ...t\{ \left( V^\prime(r,t) - \frac{1}{\beta r} \right) \rho(r,t) \right\} \right]$$ (16)

This, in turn, can be reduced to the form of Eq. (7) by introducing the effective one-dimensional potential $V_$eff$(r,t) \equiv V(r - f(t)) - \beta^{-1}\, \ln r$. The rest of the analysis follows by analogy to the linear three-state ratchet.

Figure 6: Fractionation in a radial optical thermal ratchet. (a) Pattern of concentric circular manifolds with $L = 4.7~\ensuremath{\unit{\mu m}}\xspace$. (b) A mixture of large and small particles interacting with a fixed trapping pattern. (c) Small particles collected and large excluded at $L = 4.9~\ensuremath{\unit{\mu m}}\xspace$ and $T = 4.5~\unit{sec}$. (d) Large particles concentrated at $L = 5.3~\ensuremath{\unit{\mu m}}\xspace$ and $T = 4.5~\unit{sec}$. The scale bar indicates 10  $\unit{\mu m}$.

Like the linear variant, the three-state radial ratchet has a deterministic operating regime in which objects are clocked inward or outward depending on the sequence of states (29). The additional geometric term in $V_$eff$(r)$ and the constraint that $r > 0$ substantially affect the radial ratchet's operation in the stochastic regime by inducing a position-dependent outward drift. In particular, a particle being drawn inward by the ratchet effect must come to a rest at a radius where the ratchet-induced flux is balanced by the geometric drift. Outward-driven particles, by contrast, are excluded by the radial ratchet. Combining this effect with the three-state ratchet's natural propensity for mobility-dependent flux reversal suggests that radial ratchet protocols can be designed to sort mixtures in the field of view, expelling the unwanted fraction and concentrating the target fraction. This behavior is successfully demonstrated in Figs. 6(c), in which 1  $\unit{\mu m}$ diameter silica spheres (Bangs Laboratories, lot number 21024) have been collected within an outward-driving radial ratchet at $L = 4.9~\ensuremath{\unit{\mu m}}\xspace$ at $T = 4.5~\unit{sec}$, while larger 1.53  $\unit{\mu m}$ diameter silica spheres are expelled, and in Fig. 6(d), in which the opposite is achieved with an inward-driving ratchet at $L = 5.3~\ensuremath{\unit{\mu m}}\xspace$ and the same period, $T = 4.5~\unit{sec}$. A larger and more refined version might sort different fractions into concentric rings within the ratchet domain. This capability might find applications in isolating and identifying individual bacterial species within biofilms, for example.