Pierre-Yves
Gires‡
,
Mithun
Thampi‡
and
Matthias
Weiss
*

Experimental Physics I, University of Bayreuth, Universitätsstr. 30, D-95447 Bayreuth, Germany. E-mail: matthias.weiss@uni-bayreuth.de

Received
7th July 2020
, Accepted 11th September 2020

First published on 11th September 2020

Mixing of reactants in microdroplets predominantly relies on diffusional motion due to small Reynolds numbers and the resulting absence of turbulent flows. Enhancing diffusion in microdroplets by an auxiliary noise source is therefore a topical problem. Here we report on how the diffusional motion of tracer beads is enhanced upon agitating the surrounding aqueous fluid with miniaturized magnetic stir bars that are compatible with microdroplets and microfluidic devices. Using single-particle tracking, we demonstrate via a broad palette of measures that local stirring of the fluid at different frequencies leads to an enhanced but apparently normal and homogenous diffusion process, i.e. diffusional steps follow the anticipated Gaussian distribution and no ballistic motion is observed whereas diffusion coefficients are significantly increased. The signature of stirring is, however, visible in the power-spectral density and in the velocity autocorrelation function of trajectories. Our data therefore demonstrate that diffusive mixing can be locally enhanced with miniaturized stir bars while only moderately affecting the ambient noise properties. The latter may also facilitate the controlled addition of nonequilibrium noise to complex fluids in future applications.

Despite the frequent use of microdroplets, an inherent limitation of such tiny sample volumes is often a poor, diffusion-based mixing since low Reynolds numbers prohibit any turbulence. A typical flow velocity of v = 100 μm s^{−1} for water in a microfluidic device with a typical dimension of L = 50 μm yields, for example, Re = ρvL/η < 0.01. The remaining laminar flow is hence well described by the Stokes equation and mixing of reactants, e.g. enzymes and substrate proteins, solely relies on diffusion. Making matters worse, diffusion in complex fluids, e.g. in cell extract droplets, is often slowed down or even anomalous due to (macromolecular) crowding.^{13,14} As a result, encounter rates for interaction partners are even further decreased, hence limiting the progress and timing of desired reactions. Facilitating diffusion processes in microdroplets by adding auxiliary fluctuations in a controlled fashion is therefore of high interest. Besides such a direct technical benefit, adding controlled active noise to complex fluids is also a promising tool to further probe predictions from stochastic thermodynamics^{15,16} or to study noise-dependent changes in emergent phenomena, e.g. for the self-assembly of a mitotic spindle apparatus inside a microdroplet.^{11}

Here we report on how the diffusional motion of tracer beads in aqueous microdroplets is enhanced upon agitating the fluid with micron-sized magnetic stir bars. In particular, using single-particle tracking data (SPT), we demonstrate via a broad palette of measures that locally stirring the fluid with an ensemble of stir bars at different frequencies leads to an enhanced but apparently normal and homogenous diffusion process of immersed beads: diffusional steps follow the expected Gaussian distribution and no ballistic or superdiffusive motion is observed, yet diffusion coefficients are increased up to twofold. Signatures of the local stirring are, however, clearly visible in the power-spectral density and in the velocity autocorrelation function of trajectories, indicating the addition of an auxiliary, non-white noise. Our data therefore demonstrate that local diffusive mixing can be enhanced with miniaturized stir bars while only moderately affecting the ambient noise properties. We envisage our stirring approach to be a versatile tool for microfluidic applications in which flow-driven mixing is impractical, e.g. for isolated microdroplets. In addition, the observation of only modest changes in the diffusion characteristics may be of significance for studying nonequilibrium noise under well-defined conditions.

Starting from purified SBs, the sample was produced by mixing 8 μl of the stir-bar suspension with 1.5 μl water and 0.5 μl of a suspension of fluorescent beads (200 nm FluoSpheres carboxylate, yellow-green 505/515, 2% solid from Invitrogen; diluted to a ratio 1:10^{4} and sonicated for 15 min). For each experiment, 0.4 μl of this solution was pipetted into a 4 μl squalene droplet between two 15 mm coverslips using a spacer made from four layers of magic tape (thickness of 50 μm each) and one double side tape (height of 87.5 μm). The sandwiched sample was then mounted on a custom-made aluminium holder and positioned in the center of two pairs of custom-made Helmholtz coils with which SBs were addressed (see ref. 17 for all relevant details). A custom-made microcontroller enabled us to tune the alternating magnetic field in the frequency range f ∈ [0,10] Hz with an adjustable magnetic field strength (here chosen to be 0.5 mT). A schematic representation of the resulting sample is shown in Fig. 1.

Fig. 1 Schematic representation of an aqueous droplet with magnetic stir bars and fluorescent beads (sample thickness h = 290 μm, diameter d = 620 μm). Due to an alternating magnetic field B, stir bars undergo a synchronous rotation with angular frequency ω_{B} while fluorescent beads show a vivid diffusional motion (see supplementary movies in the ESI,† for an example). |

Images of moving fluorescent beads were analyzed with FIJI. Images were first subjected to a background correction (subtraction of the mean pixel intensity from a homogeneous region around the droplet). Then, for improved particle detection, a Gaussian filter (radius 0.65 pixels) was applied and the contrast was adjusted so that 0.3% of all pixels were saturated. These images were then evaluated with the FIJI plugin TrackMate using a blob diameter of 3.6 μm, a threshold of 15 and a median filter. The maximum linking distance and gap closing were set to 5 μm with zero frame gap. If aggregates were observed (by visual inspection), these were removed from the list of trajectories. For each frequency at least 246 trajectories, each with a minimum of N = 200 positions, were used for further analysis. For each condition, trajectories were cut to the same length for all analyses.

Particles were tracked in the absence of any stirring (f = 0 Hz) and while stir bars were agitating the fluid at a stirring frequency in the range f ∈ [0.01,10] Hz. For each frequency, more than 200 trajectories were acquired and analyzed. In particular, for each trajectory r(t) with N ≥ 200 positions and a sampling time Δt = 118 ms we first calculated the time-averaged mean square displacement (TA-MSD)

(1) |

In general, a scaling law

〈r^{2}(τ)〉 = Kτ^{α} | (2) |

Using eqn (2), we have extracted K and α from individual TA-MSDs by linear regression of log(〈r^{2}(τ)〉_{t}) vs. log(τ). Since eqn (1) relies on finite averages at every lag time τ, we have restricted this fit process to τ ≤ 10Δt; for larger lag times, growing statistical fluctuations due to insufficient averaging emerge that eventually can spoil the fit process. Therefore, parameters derived from TA-MSDs essentially report on short-time transport characteristics. As a result, we obtained for each stirring frequency f the experimental probability density functions (PDFs) p(α) and p(K) for the associated ensemble of trajectories. Mean values of these PDFs are named 〈α〉 and 〈K〉 in the following. Due to the restriction of the fitting procedure to early lag times, 〈α〉 and 〈K〉 encode only the short-term characteristic of trajectories. As a complement, we therefore also fitted for each stirring frequency the ensemble-averaged TA-MSD, 〈r^{2}(τ)〉_{t,E}. Here, the fitting process included all available lag times since the ensemble averaging smoothed out the fluctuations of individual TA-MSDs. The resulting parameters, named α_{e} and K_{e}, hence encode the characteristic features over all available time scales.

In the absence of stirring (f = 0 Hz), TA-MSDs and their ensemble average showed the anticipated normal diffusion behavior 〈r^{2}(τ)〉 ∼ τ (Fig. 2a) with individual TA-MSDs exhibiting significant fluctuations around the mean scaling due to the finite time series length.^{18} These fluctuations resulted in random variations of individual scaling exponents, reflected in the width of p(α) around its mean, 〈α〉 = 1.03 (Fig. 2a, inset). Analyzing the associated 〈r^{2}(τ)〉_{t,E} yielded α_{e} = 1.02, i.e. both measures are in very good agreement with the expected value α = 1. It is worth noting that minor deviations from unity are frequently observed in SPT experiments (see discussion in ref. 18 and 19).

Fig. 2 (a) Representative TA-MSDs of fluoresecent beads, 〈r^{2}(τ)〉_{t} (grey lines), and the ensemble average of all TA-MSDs, 〈r^{2}(τ)〉_{t,E} (black circles), show normal diffusion (indicated by black dashed line) when no stirring is applied (f = 0 Hz). Inset: Fitting individual TA-MSDs with eqn (2) yields a fairly narrow probability density function of scaling exponents, p(α), with a mean 〈α〉 = 1.03, consistent with normal diffusion. The width of p(α), i.e. the standard deviation σ ≈ 0.08, is mostly determined by the finite length of trajectories (cf. discussion in ref. 18). (b) Representative ensemble-averaged TA-MSDs, 〈r^{2}(τ)〉_{t,E}, at the indicated stirring frequencies all show an approximately normal diffusive behavior, yet with varying prefactors K_{e}. Especially for f = 1 Hz, also a stirring-induced bump is visible. See main text for a detailed discussion. |

Being consistent with normal diffusion, the generalized transport coefficient for the acquired ensemble of two-dimensional trajectories was expected to be K = 4D_{0} = 9.2 μm^{2} s^{−1}, where D_{0} ≈ 2.3 μm^{2} s^{−1} is the fluorescent beads' diffusion constant in water at room temperature according to the Einstein–Stokes equation. This expectation is nicely confirmed by the experimentally determined values 〈K〉 ≈ 9.8 μm^{2} s^{−1} and K_{e} ≈ 9.5 μm^{2} s^{−1}. Again, residual uncertainties on the precise value of K are frequently observed in several measurement techniques, including SPT.^{19}

While the data for unstirred fluids confirmed, as expected, a normal diffusion with the predicted diffusion constant, any stirring action of stir bars was initially expected to enforce a superdiffusive or even ballistic signature in the MSD (1 < α ≤ 2), at least for short and possibly intermediate time scales. The observation of a synchronous rotation of stir bars (ESI,† movie 1) even enforces this expectation. Yet, upon stirring the system, all curves for 〈r^{2}(τ)〉_{t,E} retained an almost linear scaling (Fig. 2b) albeit with an elevated prefactor, K_{e}. Notably, the variation of K_{e} was non-monotonous as 〈r^{2}(τ)〉_{t,E} for f = 5 Hz assumed values between the respective curves for f = 0 Hz and f = 1 Hz. Moreover, a pronounced bump in 〈r^{2}(τ)〉_{t,E} at early lag times appeared for f = 1 Hz due to the stirring.

To explore this in more detail, we next determined the mean scaling exponents and transport coefficients at each stirring frequency (i) via the PDFs p(α) and p(K) of TA-MSDs, yielding 〈α〉 and 〈K〉, and (ii) by fitting the ensemble-averaged MSD, yielding α_{e} and K_{e}. As a result, we observed that α_{e} only showed minor deviations from unity for all driving frequencies whereas 〈α〉 showed a significant excursion for intermediate values of f (Fig. 3a). This excursion of the short-time scaling exponent 〈α〉 is directly connected to the aforementioned visible bump in MSDs (cf.Fig. 2b), whose position τ_{B} was seen to migrate towards smaller lag times as τ_{B} ∼ 1/f for increasing f.

Fig. 3 (a) In agreement with the qualitative impression from Fig. 2b, scaling exponents α_{e} from 〈r^{2}(τ)〉_{t,E} (black circles) only show minor deviations from unity for all stirring frequencies. In contrast, extracting 〈α〉 from the PDF of individual TA-MSDs (red open diamonds) highlights a significant excursion at f ≈ 1 Hz. This excursion is related to a progressive shift of the bump in MSDs towards shorter lag times for increasing driving frequencies to which fitting of TA-MSDs is more sensitive than the ensemble-averaged 〈r^{2}(τ)〉_{t,E}. (b) Associated generalized transport coefficients, K_{e} (black circles) and 〈K〉 (red open diamonds), normalized to the respective values without stirring (denoted by superscript ‘(0)’), both show an almost twofold increase for intermediate driving frequencies that subsides for larger frequencies. The anticipated linear growth 〈K〉 ∼ f is well reproduced in the range f < 0.5 Hz (full red line, y = 1 +f with y = 〈K〉/〈K^{(0)}〉 and 〈K^{(0)}〉 denoting the value at f = 0 Hz). For frequencies f ≥ 0.6 Hz a power-law decrease due to tilting of stir bars (y = 1 + 0.5/f, red dashed line) is also in qualitative agreement with the expectations. See main text for details. Error bars indicate the standard deviation of the mean for 〈α〉 and 〈K〉, and the 95% confidence bounds of the fitting parameters α_{e} and K_{e}. |

Complementing this analysis, we also calculated the temporally varying apparent scaling parameter of the ensemble-averaged TA-MSDs, defined by α(τ) = dln〈r^{2}(τ)〉_{t,E}/dlnτ. Please note that any meaningful power-law scaling needs to be defined over at least one order of magnitude, i.e. local variations in α(τ) rather reflect fluctuations that are superimposed to the general scaling exponent α_{e} shown in Fig. 3a. Following previous work,^{20} we approximated the logarithmic derivative by a polynomial to obtain a smoothly varying function α(τ). As expected from our above analysis, marked fluctuations around a mean of approximately unity were visible in α(τ), especially for driving frequencies in the range f ∼ 1 Hz (see representative data in Fig. S1 in the ESI†). These local variations, superimposed to the mean scaling exponent α_{e}, arise mostly from the transient bump in MSDs (cf.Fig. 2b). Although one might expect a superdiffusive motion for very short time scales, neither MSDs nor α(τ) revealed consistent signatures for this, supposedly due to a too poor temporal resolution Δt.

For the generalized transport coefficients we also observed significantly increased values for intermediate values of f, this time for both, 〈K〉 and K_{e} (Fig. 3b). As compared to the case without any stirring, an almost two-fold larger mean transport coefficient was seen for intermediate stirring frequencies which eventually subsided for large f. Hence, despite an almost synchronous rotation of stir bars, the apparent mode of motion stayed roughly normally diffusive, but the apparent diffusion constant was significantly elevated due to the ambient stirring at intermediate driving frequencies. Considering the fluctuation–dissipation theorem, i.e. the Einstein–Stokes equation D = k_{B}T/6πηR, this experimental finding indicates that the stirring adds an ambient active (and non-white) noise with an energy of the order k_{B}T to the mere thermal motion of beads. The effective noise temperature experienced by beads is therefore up to twice the thermal energy.

The non-monotonous variation of 〈K〉 and K_{e} is at first glance surprising. However, inspecting the motion of stir bars in more detail reveals the underlying reason: balancing the magnetic and friction-induced torques leads to a progressive tilting of the rod-like stir bars with respect to the imaging plane for f > 0.5 Hz at the used magnetic field strength.^{17} Bearing in mind this experimental finding, and following previous arguments^{21} that consider only the mean distance between tracers and adjacent vortices (i.e. stir bars), one can obtain an approximate prediction of how 〈K〉 should vary with f by considering measured values of K to be proportional to the Peclet number,

(3) |

Here, f = 1/T while _{p} denotes the stir bar's projected length in the imaging plane and D_{0} is the beads' thermal diffusion constant without any driving. For low driving frequencies, no tilting occurs and _{p} is simply the constant total length of the stir bar. Therefore, K ∼ P ∼ f is expected for frequencies f ≤ 0.5 Hz, which matches well the experimental data (see Fig. 3b). For larger frequencies, stir-bar tilting leads to a reduction of _{p}. Approximating the decrease as _{p} ∼ 1/f for f ≥ 0.5 Hz predicts K ∼ P ∼ f_{p}^{2}∼ 1/f, which agrees qualitatively with the experimental data for 〈K〉 (Fig. 3b).

Aiming to retrieve even more information from MSDs, we also probed the so-called ‘ergodicity breaking parameter’,^{22,23}E(τ) = 〈〈r^{2}(τ)〉_{t}^{2}〉_{E}/〈r^{2}(τ)〉_{t,E}^{2} − 1, that summarizes how much individual TA-MSDs fluctuate around their ensemble mean. For normal Brownian motion, E(τ) = 4τ/(3NΔt) is analytically predicted, whereas simulations reveal some finite-size deviations towards slightly larger values (ESI,† Fig. S2). Without stirring, our experimental data were in favorable agreement with this expected behavior (ESI,† Fig. S2); minor deviations for short lag times supposedly are remnants of a residual dynamic localization error.^{24} Upon stirring with a frequency in the range f ∼ 1 Hz, however, marked deviations over almost the entire range of lag times emerged (ESI,† Fig. S2). Similar to transport in mucin hydrogels,^{25}E(τ) assumed consistently larger values than predicted for normal diffusion at thermal equilibrium. Following this earlier report, we have used an empirical fit of the form E(τ) ∼τ^{ε} to quantify these deviations. As a result, we observed that very low and very high stirring frequencies featured ε ≈ 1, whereas the intermediate regime f ∼ 1 Hz, in which an enhanced diffusion was observed, yielded values ε ≈ 0.7 (ESI,† inset of Fig. S2). Therefore, stirring at intermediate frequencies induces a markedly higher spreading of individual TA-MSDs with respect to their ensemble average.

To further explore to which extent the apparently normally diffusive, but still driven trajectories deviate from mere thermal processes, we next inspected the PDF of normalized step increments. Previous reports have highlighted that spatiotemporally heterogeneous random walks, e.g. an intermittent switching between faster and slower diffusion (near and far away from a rotating stir bar), lead to characteristic signatures in this quantity,^{26–30} suggesting that the stirring could be detectable in this measure. We therefore calculated for each trajectory the time series of successive increments, Δx_{i} = x_{i+1} − x_{i} and Δy_{i} = y_{i+1} − y_{i}, and normalized these by their respective root-mean-square step length. For a homogenous normal random walk, the PDF of these normalized increments, p(χ), should follow a standard Gaussian. Since we did not observe systematic differences between x- and y-directions nor did we observe any asymmetry between positive and negative increment values, we combined all normalized increments into a single set and only considered their modulus. The resulting PDF, p(|χ|), also known as van Hove function,^{13} did not show any significant deviations from the expected Gaussian for any driving frequency (Fig. 4a). Performing the same analysis for increments taken within periods 2Δt, 3Δt etc. also did not show consistent deviations from a Gaussian (apart from the anticipated lower statistics). Nonetheless, it is tempting to speculate that image acquisition with higher frame rates, combined with a massive increase of trajectory data, could reveal diffusion heterogeneities that have been predicted for random-diffusivity models.^{31,32}

Fig. 4 (a) The PDF of the moduli of normalized increments, p(|χ|), follows the anticipated standard Gaussian (indicated by grey area) for all driving frequencies (color-coded). Therefore, no significant diffusional heterogeneity is observed due to stirring. (b) The autocorrelation of normalized squared increments, G(τ), hardly deviates from zero for all driving frequencies (color-coded). Comparing these experimental data to simulations (see ESI†) suggests that particles experience a continuously varying, rather than a dichotomously switching, mobility along their (random) trajectory in the array of rotating stir bars. See main text for more details and discussion. |

To complement the analysis via p(|χ|) and to explore in more detail whether stirring induces a varying mobility like in random-diffusivity models, we analyzed the ensemble-averaged autocorrelation function of fluctuations in the squared increments Δr^{2}(t) = |r(t + Δt) − r(t)|^{2}, defined as

(4) |

If there is a change between two transport processes, a decay G(τ) ∼ exp(−λτ) is expected, with the characteristic rate λ being determined by the switching rates between the different modes of motion.^{29,33} A meaningful amplitude of the autocorrelation function is given by G(2Δt) since even homogenous random walks yield a nonzero value for G(Δt).^{29} Inspecting G(τ) for our experimental data revealed an almost vanishing amplitude of the autocorrelation function for all stirring frequencies and lag times (Fig. 4b). Moreover, the low values of G(τ) were plagued by strong fluctuations, which were even stronger when using individual trajectories. Disregarding the fluctuations in G(τ), still a slight dependence of the small but nonzero correlation amplitude on the stirring frequency was observable. We hypothesized that this can be explained by assuming particles to undergo a simple thermally driven diffusion far away from stir bars while diffusing with an enhanced mobility when stochastically entering the vicinity of a rotating stir bar. The enhancement factor for the mobility would then be determined by the stirring frequency as shown in Fig. 3b. Performing simulations along these lines (see ESI†), we observed indeed that the autocorrelation amplitude increases with the enhancement factor. Yet, a dichotomous stochastic switching between two diffusion coefficients resulted in amplitudes of G(τ) that were at least fivefold larger than the experimentally observed ones (ESI,† Fig. S3). However, allowing for a continuously varying diffusion coefficient resulted in markedly lower amplitudes that were consistent with our experimental data (ESI,† Fig. S3). Thus, our experimental trajectories most likely do not encode a dichotomous stochastic switching between just two different modes of motion (e.g. far away and near to a stir bar), but rather they report on a continuously varying mobility along the (random) trajectory in the array of rotating stir bars.

As an additional step, we have also probed the power-spectral density (PSD) of individual trajectories, defined as

(5) |

Having observed that the PSD can capture at least part of the active stirring, we finally also inspected whether the normalized, ensemble- and time-averaged velocity autocorrelation function (VACF) is sensitive to the additional noise that is introduced by locally stirring the fluid. Being defined as

(6) |

Fig. 6 Representative VACF curves [eqn (6) with δt = 5Δt] without stirring (black circles) and with stirring at the indicated frequencies (blue squares and red diamonds). As expected, no memory is observed in the VACF for the unstirred case (i.e. zero correlation for τ > 5Δt) whereas oscillatory correlations are observed upon stirring. Inset: The VACFs oscillation frequencies ω_{0} = 2π/T are in excellent agreement with the stir bars' angular driving frequencies ω = 2πf in the range 1 rad s^{−1} ≤ ω ≤ 20 rad s^{−1}. |

In summary, we have reported here on how rotating miniaturized magnetic stir bars, embedded in aqueous microdroplets, can enhance the local diffusion of mesoscopic objects with a typical size on the 100 nm scale. Single-particle tracking on fluorescent beads allowed us to reveal that the diffusion characteristics remained almost normal, yet with an elevated diffusion coefficient and clear signatures of the stirring process in the power-spectral density and in the velocity autocorrelation function. At first glance, tracer motion in the array of almost synchronously rotating stir bars bears some similarities to reciprocal swimmers.^{37} Yet, for reciprocal swimmers the fluid experiences no net force and torque, whereas stir bars in our experiments clearly induce rotlets throughout the fluid, i.e. particles stochastically explore an array of vortices that locally accelerate them.

Due to the onset of an out-of-plane precessing of the stir bars (in our example at a stirring frequency f ∼ 1 Hz), only a twofold enhancement of diffusive transport could be achieved. The effective noise temperature applied to the fluid and picked up by the beads was therefore only up to twice the actual thermal temperature. It is worth emphasizing here that the introduced noise is by far not uncorrelated and white since our stir bars rotated almost synchronously. A simple transfer to the classical fluctuation–dissipation relation and naming a meaningful effective noise temperature on this basis is therefore only a crude way of characterizing the noise. This caveat may be addressed by decorrelating stirrers to achieve a turbulent state (see ref. 38 for an example).

In future applications, such as chemical synthesis in microdroplet reactors or in agitated non-equilibrium systems, a further enhancement of the introduced noise, besides a potentially desired decorrelation of stirrers, may be achieved by applying higher magnetic field strengths since the instability onset scales linearly with this parameter.^{39} Thus, a further enhancement of mixing by agitated diffusion in microdroplets and/or a controlled addition of nonequilibrium fluctuations with a tunable range are possible.

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## Footnotes |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/d0cp03629c |

‡ These two authors contributed equally. |

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