Line of Gaussian Wells

After convolution with a Gaussian form factor, a single well of intrinsic width $w$ takes the form

$$V_1(\vec r) = - V_0 \, \exp \left( - \frac{r^2}{2 \sigma^2} \right)$$ (60)

with $\sigma^2(a) = w^2 + a^2$. This should not be mistaken for an accurate model of an optical tweezer's potential well, but rather as a tractable model whose behavior approximates that observed in actual optical traps. A line of such wells separated by a distance $b$ results in the potential energy landscape

$$V(\vec r) = V_0(a) \sum_{n} \exp\left( - \frac{(\vec r - n b \hat x)^2}{2 \sigma^2(a)} \right).$$ (61)

Any trajectory locked in to this periodic landscape will itself be periodic in $x$. This means that such a trajectory passes through a sequence of turning points at which $\partial_x y(x,t) = \partial_t y(x,t) = 0$. Any trajectory lacking such turning points cannot be locked in, and so must escape from the line of potential wells. Turning points come in two varieties: those where particles make their nearest approaches to the wells' centers, and those corresponding to their furthest excursions from the line of traps. Particles can escape when the latter type disappear.

For small to moderate driving angles $\theta$, the more distant turning points occur near the midplanes between traps, where the restoring force is weakest. Considering the influence of just two traps (appropriate for $b > \sigma$), centered at $x = 0$ and $x = b$, this suggests the point of escape will be near $x = b/2$ and $y = \sigma$. Expanding around this point yields

$$\sin \theta_m \lesssim \eta(a) \, \exp\left( - \frac{b^2}{8 \sigma^2} \right).$$ (62)

where $\eta(a) = (2/\sqrt{e}) \, V_0/( \sigma F_0)$ measures the traps' strength relative to the driving force. Because Eq. (63) is an upper bound, no locked-in trajectories can occur for $\theta > \theta_m$. Equation (63) therefore establishes the exponential size dependence of particles' deflection.

Figure 6 shows results of numerical simulations of transport across a line of Gaussian potential wells. These simulations were designed to model the experimental design of Ref. (6), in which colloidal spheres are driven by flowing fluid past an inclined line of discrete optical traps. The driving force for this system is $F_0 \approx \xi a u$, where $\xi$ is the viscous drag coefficient corrected for hydrodynamic coupling to walls, $a$ is the radius, and $u$ is the flow speed. The sample trajectories in Fig. 6(a) were calculated for $w = 0.4 \, b$ and $\eta(a) = 9.7 \, \sigma^2/a^2$ at a fixed orientation of $\theta = 17.5^\circ$. The demonstrate that spheres with radii larger than $a = 0.1\, b$ are locked in to the array of traps under these conditions, while smaller spheres escape. Even a comparatively short array can resolve differences in radius of just a few percent, suggesting that nanometer-scale resolution should be attainable for hundred-nanometer-scale spheres in practical optical implementations.

Figure 6(b) shows how the marginally locked-in angle varies with size for this array. The lower dashed curve in Fig. 6 is the prediction of Eq. (63). Its very good agreement with simulation results in this parameter range confirms that the limiting argument establishes a useful lower bound on the $\theta _m$. These results therefore confirm that fractionation by a line of traps offers exceptional size selectivity in an appropriate range of conditions. Figure 6(b) also demonstrates that the locked-in fraction can be deflected to large angles, contrary to assertions in previous reports (5).

Figure 6: (a) Trajectories calculated according to Eqs. (54) and (55) for the line of Gaussian wells described by Eq. (62). The wells are separated by distance $b$ and have intrinsic width $w = 0.4 \, b$. Their effective width is $\sigma = \sqrt {w^2 + a^2}$, where $a$ is the radius of a sphere flowing through the array. The effective potential well depth is $\eta(a) = (2/\sqrt{e}) \, V_0/(\sigma F_0) = 9.7 \,\sigma^2/a^2$. With the driving force oriented at $\theta = 17.5^\circ$, spheres with radii larger than $a = 0.1~b$ are locked into the line. (b) Dependence of the marginally locked-in deflection angle $\theta _m$ on radius, $a$. The lower dashed curve is the prediction of Eq. (63) and the upper from Eq. (60). The dashed line indicates the orientation along which the data in (a) were calculated.

While $b > \sigma$ ensures optical fractionation's exponential size selectivity, other considerations provide a basis for optimizing the inter-trap separation. The total lateral deflection for a captured particle in an $N$-trap array is $(N-1) \, b \, \sin \theta$. The array's efficiency can be defined accordingly as the lateral deflection per trap: $\Delta(a,b) = b \sin \theta.$ Choosing $b = 2 \sigma(a)$ optimizes this efficiency at $\Delta = (4/e) \, V_0/F_j$. This result, however, does not necessarily optimize sensitivity to particle size.

The sensitivity may be formulated as

$$S(a,b) \equiv \frac{\partial \Delta(a,b)}{\partial a},$$ (63)

and is optimized by setting

$$\frac{\partial S(a,b)}{\partial b} = \frac{\partial^2 \Delta(a,b)}{\partial b \partial a} = 0.$$ (64)

This yields an optimal separation somewhat larger than that for maximum deflection:

$$\frac{b^2}{4 \sigma^2} = 1 + \chi(a) + \sqrt{ 3 + \chi^2(a)},$$ (65)

with

$$\chi(a) = \frac{1}{2}\,\left[1 - \frac{\eta^\prime(a)}{\eta(a)}\,\frac{\sigma(a)}{\sigma^\prime(a)}\right].$$ (66)

Although fractionation by a line of optical traps has been demonstrated in practice (6), optimization based on these criteria has yet to be implemented.