Deterministic Limit

A particle's trajectory through the sinusoidal landscape is described in the deterministic limit by Eqs. (19) and (20), with

$$F_y(y) = \alpha \sqrt{2\pi} I_0 k_0 a^3 \, \exp\left(- \frac{1}{2} \, k_0^2 a^2\right) \, \sin (k_0 y).$$ (30)

Non-separable form factors again yield similar results, because $I(\ensuremath{{\mathbf{r}}}\xspace )$ is independent of $x$. Because $\sin(k_0 y) \le 1$, particles become locked into the $\hat x$ direction for orientations satisfying

$$\sin \theta \leq \sin \theta_m = \frac{\lvert \alpha \rvert \, I_0}{F_0} \, \sqrt{2 \pi} k_0 a^3 \, \exp\left(- \frac{1}{2} \, k_0^2 a^2\right).$$ (31)

By contrast to Eq. (23), this reflects an exceptional sorting sensitivity: whether or not a particle becomes entrained by the fringes depends exponentially on particle size for $k_0 a > 1$. Among established fractionation schemes, only affinity chromatography offers comparable selectivity (8), and this can operate only on discrete samples of a limited class of macromolecules. Fractionation in a sinusoidal landscape, by contrast, can operate on continuous sample streams and can be implemented for a wide range of sample types.

Equations (19), (20), and (30) can be directly integrated for this simple landscape. The motion in the $\hat x$ direction is trivial:

$$x(t) = v_0 \cos \theta \, t .$$ (32)

If the particles are locked in (i.e., if Eq. (32) is satisfied), then the particles make no progress in the $\hat y$ direction, and $y(t)$ is constant in steady state. Otherwise, integration gives

$$y(t) = \frac{2}{k_0} \, \arctan\left[ \sqrt{\frac{\sin\theta + \e... ...\tan \left( \frac{k_0 v_0 t}{2} \, \sqrt{\sin^2\theta - \eta^2}\right) \right],$$ (33)

where the relative strength of the landscape's modulation is

$$\eta(a) = \frac{I_0 \, k_0 \tilde f(0, k_0a)}{F_0}.$$ (34)

The time required to advance one fringe spacing, $b = 2 \pi / k_0$, can be seen to be

$$T = \frac{2 \pi}{k_0 v_0 \sqrt{\sin^2 \theta - \eta^2}},$$ (35)

which yields a mean velocity in the $\hat y$ direction of

$$\langle v_y \rangle = v_0 \sqrt{\sin^2\theta - \eta^2}.$$ (36)

On average, the particle travels at an angle $\psi$ to the $\hat x$ axis, given by

$$\tan \psi = \frac{\langle v_y\rangle}{\langle v_x \rangle} = \beg... ...\sqrt{\sin^2\theta-\eta^2}}{\cos\theta} , & \sin \theta > \eta \end{cases} \, .$$ (37)

Figure 2: (a) Travel direction as a function of orientation for an inclined sinusoidal landscape as a function of orientation for fixed size and $\eta (a) = 0.1$, 0.2, and 0.3. Trajectories are locked in to $\psi (\theta ) = 0$ for $\theta \le \theta _m$. The diagonal dashed line indicates the result with no landscape. (b) Deflection angle as a function of particle size $a$ at fixed driving orientation $\tan \theta = 0.441$, assuming $\eta _0 = 0.4$, independent of $a$.

The deflection angle is plotted in Fig. 2(a) as a function of the driving force's orientation $\theta$ for various values of the normalized potential $\eta(a)$. The direction in which particles flow increases from $\psi = 0$ as the driving force's orientation crosses the condition for marginal lock-in, $\theta_m = \arcsin \eta$. At steeper angles, $\psi$ approaches $\theta$.

While this result is quite general, we can make the dependence on particle size more explicit by assuming the following functional form for $\eta(a)$, implied by Eqs. (14) and (35):

$$\eta(a) = \eta_0 \exp \left( -\frac{1}{2} k_0^2 a^2 \right) ,$$ (38)

where $\eta_0 = 2 \pi \alpha a^3 I_0 k_0 / F_0$. Fig. 2(b) shows the resulting dependence of deflection angle on particle size, if we assume that the driving and trapping forces are adjusted such that $\eta_0$ is a constant, independent of $a$. It can be seen that particles which are not locked in to the fringes at $\psi(a) = 0$ are fanned out into various directions, depending on their size.

Unlike the case of the single fringe, where a particle either flows along the fringe or else travels in the driving direction, the sinusoidal landscape's continuous dispersion distributes heterogeneous samples into multiple fractions, but also limits the achievable size resolution. The fraction dispersed into a finite angular range $\Delta \psi$ around $\psi$ includes an associated range of sizes

$$\Delta a \approx \left(\frac{\partial \psi}{\partial a} \right)^{-1} \, \Delta \psi,$$ (39)

which, for the locked-in fraction at $\psi = \Delta \psi / 2 \ll 1$ is

$$\Delta a \approx \cos^2 \theta \, \left(\frac{\partial \eta^2}{\partial a} \right)^{-1} \, \Delta \psi^2.$$ (40)

Thus, the exponential size selectivity implied by Eq. (32) can be lost in the exponentially wide collection window imposed by $\eta(a)$ on practical implementations. This performance cannot be improved by passing the set of particles through the fringes a second time, because of the fixed relationship between $\Delta a$ and $\Delta \psi$.

Although single-stage fractionation by a sinusoidal landscape yields broad size distributions, a narrow range of particle sizes still can be captured by using the following, two-step process. The deflection angle is first set such that all particles larger than a certain size $a_2$ will be locked in. These locked-in particles are discarded, and the remaining particles are sent through a second potential landscape, with a different deflection angle, chosen such that all particles larger than a second size $a_1 < a_2$ are locked in. Only the locked-in particles from this second stage are then retained, so that all of the remaining particles have sizes in the range $\left[ a_1 , a_2 \right]$, which can be made as small as desired.

With periodicity providing the essential ingredient for achieving exponential size selectivity, it might be expected that any periodic landscape would do. Unfortunately, this is not necessarily so. We already have demonstrated in Sec. 3 that an array of well-separated Gaussian fringes offers only algebraic, rather than exponential, size selectivity. A more general periodic landscape with wavelength $2 \pi / k_0$ can be expanded as a Fourier series:

$$I(\ensuremath{{\mathbf{r}}}\xspace ) = I_0 \,\sum_{n=0}^\infty \beta_n \sin (n k_0 y),$$ (41)

with Fourier coefficients $\beta_n$. If one of these coefficients is significantly larger than all the others, then the equations of motion can be approximated by Eqs. (19) and (20), and equivalently good size selectivity will be obtained. If, on the other hand, no single component dominates, then the superposition will not necessarily perform so well.