Narrow escape problem for Brownian particles in a microsphere with internal circulation

This paper considers the narrow escape problem of a Brownian particle within a two-dimensional domain with two escape windows and an internal circulation modeled by the flow within a Hill's vortex. To account for the spatially inhomogeneous flow within the domain, a Lagrangian study is undertaken using the complete equations of motion for a dense particle which is necessary to distinguish between the various regimes as the strength of the internal circulation is varied. For very low internal circulation the particle undergoes the conventional narrow escape problem, and agreement is good with the asymptotic expression. As the internal circulation is increased, regimes are identified with different scaling for the mean escape time. The potential application of this for drug delivery (were nanoparticles are encased in a microsphere) is discussed.

Figure 1

(a) Schematic of the narrow escape problem of a Brownian particle within a microsphere (of radius $R$, traveling at speed $U$) of fluid. The Brownian particle moves according to (14) and is reflected by all boundaries except two windows, of angle $2\alpha$, on either side through which it can escape. (b) Streamlines of a Hill's vortex showing the circulatory pattern within a microsphere due to its movement through the ambient fluid.

Figure 2

Plot of variation of the nondimensional average escape time $\overline{T}D/R^2$ with $U/{\left|\mathit{v}\right|}^{\prime }$ for a particle starting at (a) $\mathit{x}=\{0,0\}$ and (b) $\mathit{x}=\{0,0.707a\}$ [point A in Fig. 1]. In Regimes II and III the scaling is ${(U/{\left|\mathit{v}\right|}^{\prime })}^{-2/3}$ and ${(U/{\left|\mathit{v}\right|}^{\prime })}^{-2}$, respectively. The symbols denote 10 nm ($\times$), 50 nm ($\square$), and 100 nm ($*$) radii particles, with the histograms of the escape time for the filled squares shown in Figs. 3– 3. The gray line is the asymptotic mean escape time with two windows and no background flow [21].

Figure 3

Histogram of the nondimensional escape time [(a)–(c)] and typical particle trajectories [(d)–(f)] for the 50 nm radius particle for $U/{\left|\mathit{v}\right|}^{\prime }=0.006$ (a, d), 0.06 (b, e), and 19 (c, f) (filled squares in Fig. 2). In panel (e) the particle has an escape time of $TD/R^2\approx 0.045$ showing the multimodal nature in Regime II. The red circle in panels (d)–(f) marks where the particle leaves the domain (between the two $\times$'s, which denote the edges of the escape window).

Figure 4

Root-mean-square displacement of a (a) 10 nm, (b) 50 nm, and (c) 100 nm particle with time, where ${\tau }_d=6\pi \mu r_p^3/\left(k_b{T}_f\right)$. The dashed line is the theoretical prediction (A2).