Perturbative theory for Brownian vortexes

Brownian vortexes are stochastic machines that use static nonconservative force fields to bias random thermal fluctuations into steadily circulating currents. The archetype for this class of systems is a colloidal sphere in an optical tweezer. Trapped near the focus of a strongly converging beam of light, the particle is displaced by random thermal kicks into the nonconservative part of the optical force field arising from radiation pressure, which then biases its diffusion. Assuming the particle remains localized within the trap, its time-averaged trajectory traces out a toroidal vortex. Unlike trivial Brownian vortexes, such as the biased Brownian pendulum, which circulate preferentially in the direction of the bias, the general Brownian vortex can change direction and even topology in response to temperature changes. Here we introduce a theory based on a perturbative expansion of the Fokker-Planck equation for weak nonconservative driving. The first-order solution takes the form of a modified Boltzmann relation and accounts for the rich phenomenology observed in experiments on micrometer-scale colloidal spheres in optical tweezers.

Figure 1

(a) Force field $\mathbf{F}\left(\mathbf{r}\right)$ experienced by a $1.0-\mu \mathrm{m}$-diameter colloidal silica sphere of refractive index 1.45 in a $1\mathrm{mW}$ Gaussian optical tweezer propagating upward along the $\widehat{z}$ axis with a ${60}^{\circ }$ convergence angle. Colors indicate relative intensity of the beam. Arrows denote direction of net force in the $(r,z)$ plane. The superimposed semicircle indicates the size of the colloidal particle. (b) Axial component of the restoring force, $F_z\left(z\right)$, along $r=0$. (c) Radial component of the restoring force, $F_r\left(r\right)$, in the plane $z=0$. Shaded rectangles in (b) and (c) indicate the particle's range of motion within the force field as a function of fluctuation energy in steps of $2k_{B}T$ up to $10k_{B}T$. (d) Nonconservative component of the force, $F_z\left(r\right)$, in the plane $z=0$. (e) Detail of (d) showing fit to a quartic polynomial.

Figure 2

Streamlines of the current density for a $1\mu \mathrm{m}$-diameter silica sphere in an optical tweezer as a function of trap intensity. Sharp ends of the symbols point in the direction of $\mathbf{j}\left(\mathbf{r}\right)$ with colors determined by the relative probability $\rho \left(\mathbf{r}\right)$, as indicated by the inset color bar. Light propagates in the $\widehat{z}$ direction. (a)–(c) show results from Brownian dynamics simulations in force fields computed for a Gaussian optical tweezer at a vacuum wavelength of $532\mathrm{nm}$ and a numerical aperture of 1.4. (d)–(f) show corresponding results of analytical expressions in Eq. (27) for the same parameters. (a), (d) $\frac{1}{2}\beta k{a}_p^2=30.5$: a single toroidal roll is visible near the optical axis. (b), (e) $\frac{1}{2}\beta k{a}_p^2=7.6$: concentric counter-rotating toroidal vortexes. (c), (f) $\frac{1}{2}\beta k{a}_p^2=3.6$: weak confinement showing a single flux-reversed roll. All of the qualitative features of the Brownian vortex circulation observed in the simulation results are obtained also in the analytical theory.

Figure 3

(a) Dependence on the particle radius of the quadratic and quartic coefficients $\epsilon$ and $p_4$ characterizing the nonconservative optical force $F_z\left(r\right)$. These coefficients are obtained by polynomial fits to the force field computed with generalized Lorenz-Mie theory for a silica sphere in a Gaussian optical tweezer. Results are shown for a silica sphere in water and an optical tweezer created with a vacuum wavelength of $532\mathrm{nm}$ in a beam with a convergence angle of ${60}^{\circ }$. (b) $F_z\left(r\right)$ for a sphere with $a_p=0.13\mu \mathrm{m}$, together with a quartic fit. (c) $a_p=0.45\mu \mathrm{m}$. (d) $a_p=0.5\mu \mathrm{m}$.