Observation of Brownian elastohydrodynamic forces acting on confined soft colloids

Significance

Confined motions of particles are ubiquitous in microbiology. Examples at the micro- and nano-scales include blood cells flowing in vessels, antibody recognition, or confined diffusion of synaptic neurotransmitters. These situations bring about viscous flows coupled to charged and soft confining entities, in presence of thermal fluctuations. Fluctuation-free scenarios have already proven the emergence of novel softness-induced near-contact forces, but the inclusion of fluctuations—crucial at microscopic scales—is yet to be explored. Here, aiming at unraveling their impact, we combine holographic microscopy and statistical inference. We observe the emergence of a piconewton-like inertia-less lift force acting on confined fluctuating soft microdroplets. This so-far-overlooked and universal type of forces could tune transient migration strategies, relevant to transport situations in microbiology.

Abstract

Confined motions in complex environments are ubiquitous in microbiology. These situations invariably involve the intricate coupling between fluid flow, soft boundaries, surface forces, and fluctuations. In the present study, such a coupling is investigated using a method combining holographic microscopy and advanced statistical inference. Specifically, the Brownian motion of soft micrometric oil droplets near rigid walls is quantitatively analyzed. All the key statistical observables are reconstructed with high precision, allowing for nanoscale resolution of local mobilities and femtonewton inference of conservative or nonconservative forces. Strikingly, the analysis reveals the existence of a novel, transient, but large, soft Brownian force. The latter might be of crucial importance for microbiological and nanophysical transport, target finding, or chemical reactions in crowded environments, and hence the whole life machinery.

An enormous number of simultaneous and overlapping processes ensure the proper functionality of the human body. Nearly all of the underlying mechanisms arise spontaneously through the passive and automated activity of the cells. Such a general statement raises questions about the fundamental mechanisms leveraged by the human-body machinery at the individual-cell scale which orchestrate greater-scale organization and operation. These mechanisms necessarily include the motility of confined microscopic entities, encountered in several situations spanning the micro- and nano-scales: from blood cells flowing in vessels and capillaries (1–3), cell differentiation, to antibody recognition, glia cells’ mechanosensing (4), and confined diffusion of synaptic receptors (5–7) or of neurotransmitters in the synaptic cleft (8). The first quest to comprehend these scenarios from a physical standpoint begins with a minimal set of ingredients: viscous flows coupled to charged and soft confining interfaces, in presence of thermal fluctuations. This reasoning yields a twofold state-of-the-art, as presented hereafter.

On the one hand, deterministic (i.e. fluctuation-free) motion in lubricated geometries (9), where a thin layer of viscous liquid separates an object from a boundary, drastically depends on the softness (i.e. ability to deform) of the entities at play (10, 11). This recent area of research, aptly named soft lubrication, revealed the appearance of surprising inertial-like forces and torques, traditionally observed for high-speed and large-scale flows. These include lift forces, Magnus-like effects, or dynamic adhesion, to name a few, which all result from the elastohydrodynamic coupling between the motion-induced viscous flow and the soft confining boundaries (12–15). Over the past decade, dedicated research has provided decisive experimental evidence for these soft-lubrication effects across a broad range of scales, from the observation of self-sustained lift and reduced friction of a macroscopic cylinder sliding down a soft incline (16), and the observation of a falling bead surfing its own elastic wave over a nearby membrane (17), to the direct and quantitative measurement of the soft-lubrication lift force at the nanoscale (18, 19). Interestingly, such interfacial elastohydrodynamic couplings play a significant role in biological contexts (20), as demonstrated by the observation of micrometric beads advected in microfluidic channels being drastically repelled from biomimetic hyaluronic acid brushes grafted on rigid walls (21, 22). A parameter not to be forgotten resides in the very nature of the interfacial softness. As the entities get softer, or equivalently at smaller scales, capillarity rather than elasticity starts to be the dominant restoring force triggering soft-lubrication couplings (23). More precisely, this shift occurs when the sizes involved in the problem become smaller than the elasto-capillary length $\gamma /E$ (24), typically $1\mathrm{\mu }\mathrm{m}$ for a material of Young’s modulus $E=10\mathrm{kPa}$ and surface tension $\gamma =10\mathrm{mN}/\mathrm{m}$, which are typical conditions easily reached both naturally and experimentally. Such capillary-induced soft-lubrication forces, coupled to static interactions, were e.g. proven to control droplet–droplet interactions (25), as well as cell–cell adhesion (26).

On the other hand, as the size of the entities decreases, Brownian (i.e. thermally fluctuating or diffusive) motion becomes prominent and is strongly affected by confinement and hydrodynamic interactions (27). Previous research, focused on rigid-confinement scenarios, already pointed out striking differences compared to the idealized bulk picture. Faucheux and Libchaber (28) demonstrated that diffusion is hindered by confining colloids between rigid walls and opened the way to several related studies (29–32). Such research efforts have not only highlighted the complex, anisotropic, and space-dependent nature of diffusion in confined spaces (33), but have also harnessed these features in order to develop novel gentle techniques for nanorheological and surface characterization (34). Besides, detailed analysis of rigid-confinement scenarios revealed that random walkers also exhibit non-Gaussian statistics of displacements (35–37), associated with the enhanced occurrence of rare events—hence with fundamental implications in winners-take-all processes for life.

Ongoing research has yet to explore the soft Brownian intersection of the two domains above, namely soft lubrication and confined Brownian motion, which remains theoretically and experimentally challenging to address. Incorporating molecular fluctuations into the nonlinear elastohydrodynamic continuum theory is inherently an arduous task. As of yet, theoretical studies remain scarce and have merely focused on the simplified case of point-sized tracers (38–40), thereby neglecting any finite-size effects and lubrication couplings. A few alternative numerical strategies have thus emerged (41–43) and tend to indicate deformability-driven anomalous diffusive behaviors. On the experimental front, evidence for soft Brownian couplings remains elusive (44), which might be attributed to the magnitude of the latter. A coarse hand-waving argument which consists in plugging the thermal root-mean-squared velocity into existing deterministic theories of soft lubrication predicts piconewton-like soft Brownian forces in typical conditions. While this order of magnitude is comparable to the one of other surface and entropic forces—hence suggesting that it might be biologically relevant in order to revisit and rationalize spontaneous migration and target-finding strategies (e.g. for proteins near cell membranes)—the soft Brownian forces that we propose here remain hypothetical and must then be measured and quantified precisely. Besides such a fundamental challenge, and although more than a century has passed since Brownian motion was first used by Jean Perrin to measure Avogadro’s number and prove the atomic hypothesis (45), the intricacies of Brownian motion at complex interfaces could further contribute to open a novel window toward ultralow force sensing in biological and nanoscale physics (45–47). Fine-tuning such technique remains imminent in the current era, where the demand for miniaturized precise tools is targeted, e.g. to probe biologically relevant elasticity-related phenomena (48).

In this article, we address both the above-mentioned fundamental and practical questions via our recently developed nonfluorescent superresolution microscopy technique based on Mie holography and advanced statistical inference (47). In the subsequent sections, we describe the methods used to generate Brownian oil microdroplets, track them, and analyze their confined motion. From the reconstruction of the whole set of statistical observables, we first robustly recover all the expected equilibrium and dynamical properties with high precision. Using such a calibrated method, we then investigate the transient emergence of piconewton-like nonconservative forces in the combined presence of soft confinement and thermal fluctuations. Mainly, our results seem to indicate the existence of a soft Brownian force, that we systematically quantify, characterize, and rationalize.

Experimental Situation

We generate and employ low-interfacial-tension oil microdroplets, with radius $a\sim 2\mathrm{\mu }\mathrm{m}$ and interfacial tension in water $\gamma =8\pm 3\mathrm{mN}/\mathrm{m}$, as detailed in Materials and Methods and SI Appendix, Table S1. We then study their random trajectories within various viscous immiscible liquids near a rigid glass substrate, as sketched in Fig. 1A.

Fig. 1.

Overview of the experimental method. (A) Schematic of the system: An oil microdroplet diffuses in a viscous liquid near a rigid glass wall. The Inset highlights the surface forces at play: electrostatic repulsion and elastohydrodynamic coupling. (B) Experimental set-up: Mie holography (47). (C) Interference patterns (i.e. holograms) for two wall-droplet distances $z$, as indicated by the symbols on the Top Right corners and corresponding ones in (D). Each two-image panel depicts, from Left to Right, the experimental hologram and the corresponding best-fit Lorenz-Mie one. (Scale bar: $5\mathrm{\mu }\mathrm{m}$.) (D) Experimental trajectory of an oil droplet (radius $a=1.6\mathrm{\mu }\mathrm{m}$) diffusing in a water-ethylene-glycol mixture (mass fraction $w=0.3$). (E) Normalized intensity profiles from the experimental and theoretical holograms for $z=0.7\mathrm{\mu }\mathrm{m}$, as functions of the radial distance to the hologram’s center. The light purple area corresponds to the SD. (F) 2D kernel density estimate of the in situ calibration of the radius $a=1.587\pm 0.016\mathrm{\mu }\mathrm{m}$ and optical index $n_{\mathrm{p}}=1.471\pm 0.002$ of the droplet used in (C and D), measured over ${10}^4$ points. (G) Sedimentation curves used as in situ calibration of medium viscosity ${\eta }_{\mathrm{m}}$ and droplet’s buoyant density $\mathrm{\Delta }\rho$. The three solid lines correspond to the used liquids: ultrapure DI water (blue, ${\eta }_{\mathrm{m}}=1\mathrm{m}Pa·s$, $\mathrm{\Delta }\rho =70\mathrm{kg}/{\mathrm{m}}^3$), 30%-mass-fraction water-ethylene-glycol mixture (purple, ${\eta }_{\mathrm{m}}=2.1\mathrm{m}Pa·s$, $\mathrm{\Delta }\rho =33\mathrm{kg}/{\mathrm{m}}^3$), and 40%-mass-fraction water-ethylene-glycol mixture (green, ${\eta }_{\mathrm{m}}=2.6\mathrm{m}Pa·s$, $\mathrm{\Delta }\rho =18\mathrm{kg}/{\mathrm{m}}^3$).

Individual microdroplets in a dilute emulsion are tridimensionally tracked through Mie holography (49), as explained in Materials and Methods and depicted in Fig. 1 B and C. Tridimensional trajectories $(x(t),y(t),z(t))$ along time $t$ are reconstructed over broad spatiotemporal ranges, as shown in Fig. 1D. This method leads to a spatial tracking resolution of $\sim$20 $\mathrm{nm}$, thanks to the strong agreement between the experimental and theoretical interference patterns (Fig. 1 C and E).

Moreover, the pattern not only depends on the droplet’s tridimensional position, but also on the droplet’s radius $a$ and optical index $n_{\mathrm{p}}$. The two latter parameters are thus measured in situ with precisions of 20 nm and 0.002, respectively, as highlighted in Fig. 1F.

The trajectories also contain information about the surrounding medium. For every experimental set, the viscosity ${\eta }_{\mathrm{m}}$ of the surrounding liquid medium and the relative buoyant density $\mathrm{\Delta }\rho$ of the droplet are also measured in situ, by studying the early-stage trajectory (Materials and Methods and SI Appendix, Fig. S1 and Table S2), i.e. the droplet’s sedimentation toward but far from the wall, thus allowing to neglect boundary effects (see Fig. 1D, $z\ge 20\mathrm{\mu }\mathrm{m}$).

The two preliminary steps described above permit to fully characterize each studied droplet, making the method entirely self-calibrated prior to the central statistical study discussed below.

Equilibrium Properties

Equilibrium Static Properties.

After sedimentation, the droplet reaches equilibrium and steadily diffuses close to the wall. It stays confined near the wall as a result of the competition between thermal fluctuations, and both the electrostatic repulsion between the negatively charged surfaces of the droplet and the wall and the droplet’s buoyant weight. This translates into Boltzmann’s equilibrium probability density function $P_{\mathrm{eq}}(z)=Nexp[-U_{\mathrm{eq}}(z)/(k_{\mathrm{B}}T)]$ of the $z$ position (Fig. 2A), where $N$ is a normalization factor, $U_{\mathrm{eq}}$ is the potential energy, $k_{\mathrm{B}}$ is Boltzmann’s constant and $T$ is the temperature. The normal conservative force $F_{\mathrm{eq}}$ acting on the droplet is then defined as:

$$\begin{array}{c}\hfill F_{\mathrm{eq}}=-\frac{\mathrm{d}{U}_{\mathrm{eq}}}{\mathrm{d}z},\end{array}$$ [1]

Fig. 2.

Equilibrium static properties for an oil droplet of radius $a=1.6\mathrm{\mu }\mathrm{m}$ diffusing in ultrapure DI water near a rigid glass wall. The red squares correspond to the experimental data. The black solid lines represent the Boltzmann factor (A) and Eq. 1 (B), that include electrostatic repulsion and the droplet’s buoyant weight. (A) Equilibrium probability distribution function $P_{\mathrm{eq}}$ as a function of the wall-droplet gap distance $z$. (B) Conservative force $F_{\mathrm{eq}}$ as a function of $z$. Error bars give a 95%-CI. The black dashed line indicates the droplet’s buoyant weight and the red-colored area the thermal-noise limit (52).

where $U_{\mathrm{eq}}/(k_{\mathrm{B}}T)=B{e}^{-z/l_{\mathrm{D}}}+z/l_{\mathrm{B}}$. While $B$ quantifies the magnitude of the surface charges, the Debye length $l_{\mathrm{D}}$ (50) and the Boltzmann length $l_{\mathrm{B}}=k_{\mathrm{B}}T/(\mathrm{\Delta }mg)$ (45) balance thermal energy with the electrostatic repulsion and the buoyant weight $\mathrm{\Delta }mg$, respectively, where $\mathrm{\Delta }m$ is the buoyant mass and $g$ the acceleration of gravity. Safety checks regarding the measurement of electrostatic properties are given in (SI Appendix, Equilibrium Dynamic Properties and Figs. S2 and S3 and Table S3), in the case of control experiments on rigid polystyrene microbeads that diffuse in salted water near a rigid glass wall. In particular, the expected values and dependencies on salt concentration for both $l_{\mathrm{D}}$ and $B$ are captured (51). As shown in Fig. 2B, femtonewton forces are measured and accurately described by Eq. 1. The error bars are of the same order of magnitude as the theoretical thermal-noise limit (52), meaning that the resolution has reached its ultimate bound for the given system and acquisition time.

Equilibrium Dynamic Properties.

A complete description of equilibrium invariably includes dynamic observables. For clarity purposes, the theoretical framework is provided in (SI Appendix). The canonical dynamic variable is the mean-squared displacement (MSD) $⟨{\delta }_{\tau }{x}_i^2⟩$ ($x_i=x,z$), shown in Fig. 3A, where ${\delta }_{\tau }{x}_i$ denotes the displacement of the coordinate $x_i$ over the time increment $\tau$, and where $⟨·⟩$ indicates a temporal average. The plot shows that, as in the bulk and rigid-confinement (47) cases, the MSDs are linear in time, at short times. The process is thus Brownian (or Fickian). The saturation observed in the vertical direction at longer times simply results from the effective trapping due to gravity. From now on, time regimes will be defined by comparison to the characteristic time ${\tau }_{\mathrm{eq}}$. The latter time scale is the ratio between the saturation value of the vertical MSD and the slope of the linear temporal regime of the MSD (see SI Appendix for the exact definitions). This time scale sets the crossover from the short-time free-diffusive regime to the long-time effectively trapped one, i.e. quantifies the time to reach equilibrium in the system. Besides, as in the rigid-confinement case, the droplet’s motion is anisotropic, resulting in a smaller average diffusion coefficient in the vertical direction as compared to the horizontal one. By binning the data in $z$ and using an advanced inference method (SI Appendix, Inference of Local Diffusion Coefficient Profiles) (53), the local diffusion coefficients can also be measured with a nanometric spatial resolution (Fig. 3B). In all cases, the measurements agree with the theoretical predictions for rigid no-slip confinement (54, 55). Furthermore, the probability density functions of the transverse and normal displacements, $P({\delta }_{\tau }x)$ and $P({\delta }_{\tau }z)$ respectively, are measured at various times (Fig. 3 C and D). Again, the experimental data match the theory established for rigid no-slip confinement (SI Appendix, Equilibrium Dynamic Properties).

Fig. 3.

Equilibrium dynamic properties for an oil droplet of radius $a=1.6\mathrm{\mu }\mathrm{m}$ diffusing in ultrapure DI water near a rigid glass wall. (A) Mean-squared displacements $⟨{\delta }_{\tau }{x}_i^2⟩$, as functions of the time increment $\tau$, for the horizontal ($x_i=x$, blue circles) and vertical ($x_i=z$, red hexagons) motions. The solid lines represent the short-time theory, and the red dashed line the long-time one (47). The vertical black dashed line indicates the equilibrium time ${\tau }_{\mathrm{eq}}$ that separates the short-time diffusive regime and the long-time effectively trapped regime. (B) Local diffusion coefficients $D_i$ (same color code), normalized by the bulk one $D_0$, as functions of the wall-droplet gap distance $z$. The solid lines represent the theory for the no-slip rigid case (47). The colored areas give 95%-CIs (56). (C) Probability density function $P({\delta }_{\tau }x)$ of the transverse displacement ${\delta }_{\tau }x$, for the two time increments $\tau$ indicated in legend. The solid lines correspond to the short-time theory (47). (D) Probability density function $P({\delta }_{\tau }z)$ of the normal displacement ${\delta }_{\tau }z$. The red solid line corresponds to the short-time theory, and the black dashed line to the long-time one (47). Error bars give a 99%-CI.

In summary, the experiments on soft microdroplets are accurately captured at equilibrium by the rigid-colloid theoretical framework, as it is the case for independent control experiments with rigid polystyrene microbeads (SI Appendix, Fig. S2). Hence, at equilibrium, a soft microdroplet behaves as its rigid counterpart. However, this statement breaks down at short time scales, as discussed hereafter.

Main Result

In addition to the conservative force obtained above from the equilibrium distribution in vertical position, one can measure from the vertical drifts the total vertical force $F_{\mathrm{tot}}$ (including conservative and possible novel nonconservative forces) in excess to the no-slip rigid-confinement Stokes-like drag, as:

$$\begin{array}{c}\hfill F_{\mathrm{tot}}(z)=\frac{k_{\mathrm{B}}T}{D_z}\left(\frac{{⟨{\delta }_{\tau }z⟩}_z}{\tau }-\frac{\mathrm{d}{D}_z}{\mathrm{d}z}\right),\end{array}$$ [2]

where the terms on the right-hand side correspond to the Stokes drag and the spurious drift (SI Appendix, Measurement of Non-Conservative Forces), and where ${⟨·⟩}_z$ denotes the average over a vertical bin. The measured $F_{\mathrm{tot}}$ contains the conservative and nonconservative contributions, except for Stokes drag by construction. This total force $F_{\mathrm{tot}}$ perfectly matches the previously mentioned conservative force $F_{\mathrm{eq}}$ in the case of rigid polystyrene spherical colloids (SI Appendix, Fig. S4), indicating that no novel nonconservative force $F_{\mathrm{nc}}=F_{\mathrm{tot}}-F_{\mathrm{eq}}$ acts on rigid colloids, as expected. Conversely, $F_{\mathrm{nc}}$ appears to be nonzero, positive, and large (i.e. in the piconewton range) as compared to the conservative force, in the case of confined soft droplets at early time scales (Fig. 4A). This observation seems to indicate the spontaneous emergence of some out-of-equilibrium vertical force. Furthermore, this force increases with the outer-medium viscosity ${\eta }_{\mathrm{m}}$ (Fig. 4B) in a superlinear, quadratic-like manner (Fig. 4C), possibly suggesting a nonlinear hydrodynamic origin. Besides, it decays as the droplet moves at larger distances from the wall, highlighting some interfacial or confinement-induced origin. Finally, this force appears to be transient, as evidenced by the temporal dependency of $F_{\mathrm{nc}}$ shown in Fig. 4D. Indeed, as the time increment $\tau$ used to evaluate $F_{\mathrm{nc}}$ increases, $F_{\mathrm{nc}}$ decreases and converges toward 0 over a few seconds (see Inset in Fig. 4D). Interestingly, this time scale is comparable to the time ${\tau }_{\mathrm{eq}}$, defined in Fig. 3A, required to reach equilibrium in the system. The Inset in Fig. 4D further suggests that such a transient nonconservative force may vanish toward equilibrium in an exponential-like manner.

Fig. 4.

Transient inertia-less visco-capillary lift force acting on microdroplets. (A) Equilibrium force $F_{\mathrm{eq}}$ (black squares; see Eq. 1), and total force $F_{\mathrm{tot}}$ (blue circles; see Eq. 2) in excess to the Stokes-like drag, as functions of the wall-droplet gap distance $z$. (B) Nonconservative force $F_{\mathrm{nc}}=F_{\mathrm{tot}}-F_{\mathrm{eq}}$ in excess to the Stokes-like drag, as a function of $z$. The black squares correspond to the case of rigid polystyrene colloids. The other colored symbols correspond to soft oil droplets in different viscous liquids. The outer-liquids’ viscosities ${\eta }_{\mathrm{m}}$ relative to the one $\eta$ of water are indicated in legend. Nonconservative forces are computed over $\tau =20\mathrm{ms}$. (C) Normalized nonconservative force ${\widehat{F}}_{\mathrm{nc}}=F_{\mathrm{nc}}(\gamma {\rho }_{\mathrm{p}}{a}^2)/({\eta }_{\mathrm{m}}^2{k}_{\mathrm{B}}T)$ as a function of the dimensionless gap $z/a$. Symbols correspond to the ones in (B). The black solid line corresponds to Eq. 4, and the gray area to the normalized thermal-noise limit (Fig. 2B). (D) Nonconservative force $F_{\mathrm{nc}}$ as a function of $z$, for different time increments $\tau$, in the case of a 2-$\mathrm{\mu }m$ droplet diffusing in a 40-% water-ethylene-glycol mixture (${\eta }_{\mathrm{m}}=2.6\eta =2.6\mathrm{m}Pa·s$). The Inset shows the average over $z$ of $F_{\mathrm{nc}}$ (in pN), as a function of $\tau$ (in s). Error bars are removed for clarity.

Discussion

In the remainder of the article, we argue that a relevant candidate to explain all these quantitative and systematic observations is the soft Brownian force proposed above. First, fluid inertia can be disregarded, as the typical inertial time ${\tau }_{\mathrm{in}}\sim {\rho }_{\mathrm{m}}{a}^2/{\eta }_{\mathrm{m}}\sim 10\mathrm{\mu }\mathrm{s}$, where ${\rho }_{\mathrm{m}}$ denotes the outer-medium density, is negligible compared to the minimal time increment $\tau =10\mathrm{ms}$ used to measure the drifts ${\delta }_{\tau }z$. Hence, the outer-fluid flow induced by the droplet motion is governed by the steady incompressible Stokes equation which, in the near-contact lubricated regime where $z\ll a$, reads ${\partial }_{r}p={\eta }_{\mathrm{m}}{\partial }_{\mathit{zz}}v$, where $p$ and $v$ respectively denote the hydrodynamic pressure and velocity fields in the fluid gap between the droplet and the wall, and $r$ is the radial horizontal coordinate. Moreover, the droplet is much more viscous than the outer medium (viscosity ratio greater than 400), so that we can neglect fluid flow inside the droplet. Carrying on with the modeling (see details in SI Appendix), we focus on the horizontal motion of the droplet with a typical velocity $V_0$. Under all these assumptions, and in the limit of small interfacial deformations $\delta$ (i.e. $\delta \ll z$), the pressure $p$ scales as:

$$\begin{array}{c}\hfill \frac{p}{l}\sim {\eta }_{\mathrm{m}}\frac{V_0}{{(z+\delta )}^2}\simeq \frac{{\eta }_{\mathrm{m}}{V}_0}{z^2}\left(1-2\frac{\delta }{z}\right),\end{array}$$ [3]

where $l=\sqrt{2az}$ is the hydrodynamic radius, i.e. the horizontal extent of the lubrication pressure field (12). The first and second terms in the right-hand side of Eq. 3 correspond to the zeroth- and first-order pressure contributions of the perturbation expansion in the deformation. The former, akin to the excess pressure field generated from the translational lubricated motion of a rigid bead near a rigid wall with no slippage (54), does not generate any net vertical force by integration because it is an antisymmetric function due to both the time-reversal symmetry of the Stokes equation and the even contact geometry. However, it deforms the droplet according to Laplace’s law $p\sim \gamma \delta /l^2$.

As a consequence of the flow-induced droplet’s deformation, the above-mentioned fore–aft contact symmetry is broken, and the first-order pressure then leads to a nonzero vertical force. After combining Laplace’s law with Eq. 3, and integrating the first-order pressure over the area $\pi l^2$, this vertical force reads $\sim {\eta }_{\mathrm{m}}^2{V}_0^2{a}^3/(\gamma z^2)$. Further using the thermal velocity $\sqrt{3k_{\mathrm{B}}T/(2\pi a^3{\rho }_{\mathrm{p}})}$ as an estimate for the horizontal velocity scale $V_0$ in the fluctuating case, and adding an ad hoc exponential cutoff factor toward equilibrium, one gets the following scaling suggestion for the soft-Brownian force:

$$\begin{array}{c}\hfill F_{\mathrm{nc}}\sim \frac{{\eta }_{\mathrm{m}}^2{k}_{\mathrm{B}}T}{\gamma {\rho }_{\mathrm{p}}{a}^2}{\left(\frac{a}{z}\right)}^{2}exp\left(-\frac{\tau }{{\tau }_{\mathrm{eq}}}\right).\end{array}$$ [4]

As observed in Fig. 4C, Eq. 4 allows to collapse all the experimental data into a single master curve for a given time lapse $\tau$. Furthermore, the correct $\sim {(z/a)}^{-2}$ power law is recovered above the thermal-noise limit of the method, with a multiplicative factor similar in order of magnitude as the one estimated theoretically (SI Appendix, Soft Lubrication Model). All together, these elements seem to indicate that the transient but large nonconservative force measured in our study does indeed emerge from an intricate coupling between softness, flow confinement, and thermal fluctuations. A final remark should nevertheless be made on the fluid’s inertia contribution to the droplet’s dynamics and resulting force. The total force measurement is performed by capturing stroboscopic glimpses of the droplet’s trajectory acquired at a frequency $1/\tau$, which is smaller than $1/{\tau }_{\mathrm{in}}$. While the measured force seems to be reminiscent of inertial dynamics, the actual droplet’s mass to consider in the velocity scale is not trivial. Indeed, fluid inertia does classically contribute to the effective mass of immersed objects in the bulk, but i) the added mass becomes space-dependent in rigid confinement (57), and ii) we might even anticipate novel softness-induced corrections to the added mass. Further experimental and theoretical investigations of fluctuating elastohydrodynamics at small time scales are thus needed in the future in order to address such a matter.

Conclusion

Using a state-of-the-art, broadband, contactless, and nonfluorescent method based on optical holography and statistical inference, we reported the observation of a transient but large spontaneous colloidal soft Brownian force, solely triggered by the coupling between thermal fluctuations, confined flow, and soft boundaries. This result is believed to be significant for nanoscience, as such a so-far-overlooked effect might strongly contribute to short-term microscopic transport, target finding, and chemical reactions in complex bounded environments—that are all ubiquitous features in microbiological and nanophysical settings.

Materials and Methods

The supporting materials for this manuscript are described at the end of SI Appendix and can be found on the online repository: https://github.com/EMetBrown-Lab/Brownian-motion-near-soft-interfaces (58).

Materials.

The oil microdroplets are made by mixing $200\mathrm{\mu }\mathrm{L}$ of oil (AR1000, purchased from Wacker©) and $10\mathrm{mL}$ of the working viscous liquid. The mixture is then put in an ultrasonic bath for ten minutes, and diluted. This procedure leads to oil droplets of radii ranging from $1\mathrm{\mu }\mathrm{m}$ to $5\mathrm{\mu }\mathrm{m}$, with high reproducibility (59, 60). Their interfacial tension $\gamma =8\pm 3\mathrm{mN}/\mathrm{m}$ in water is measured by the pendent-drop method (61). Results for different experimental conditions are presented in SI Appendix, Table S1.

Control experiments are conducted with rigid polystyrene microspheres purchased from Polysciences©. From now on, if the distinction is not necessary, the rigid polystyrene microspheres or the soft oil microdroplets will be referred to as colloids.

Chambers containing the diffusing colloids are made of one glass slide and one glass cover-slip (purchased from Academy©) and sealed thanks to melted Parafilm©.

The working viscous liquids refer to mixtures of pure deionized (DI) water (type 1, MilliQ©device) with ethylene glycol (anhydrous, 99.8%, purchased from Sigma Aldrich©). Properties of the mixtures are described in the caption of Fig. 1G of the main text and SI Appendix, Table S2.

Tridimensional Tracking of Single Colloids.

Individual micrometric colloids in a dilute suspension are tridimensionally tracked through Mie holography (49, 62). A coherent plane wave (wavelength $532\mathrm{nm}$) illuminates an isolated colloid and is thus scattered. The scattered light interferes with the incident one, leading to a pattern called hologram (Fig. 1 B and C). Holograms are recorded via an $\times$100 oil-immersed Olympus objective (numerical aperture: 1.45) for one hour at a frame rate of 100 Hz, and fitted according to the Lorenz-Mie theory thanks to the Pylorenzmie software suite (63), allowing us to reconstruct the tridimensional trajectories $(x(t),y(t),z(t))$ along time $t$, over broad spatiotemporal ranges, as shown in Fig. 1D. Experimental spatial illumination defects and variations are corrected by normalizing the image by the background image. The latter is obtained by computing the median image of the movies. Also, parameters which are independent of the colloid but affect the holograms are measured separately and fixed. The liquid’s refractive index $n_{\mathrm{m}}$ is measured thanks to a refractometer (DR-M2/1550 ATAGO©) and the results are indicated in SI Appendix, Table S2. The illumination comes from a collimated laser diode (CPS532 from Thorlabs©).

This method leads to a spatial tracking resolution of $\sim$20 $\mathrm{nm}$, thanks to the strong agreement between the experimental and theoretical holograms (Fig. 1 C and E).

In Situ Calibration.

Two preliminary steps permit to fully characterize each studied droplet, making the method entirely self-calibrated prior to the central statistical study discussed in the main text.

First, the fitting not only depends on the droplet’s tridimensional position, but also on the droplet’s radius $a$ and optical index $n_{\mathrm{p}}$. The two latter parameters are calibrated in situ by fitting ten thousand holograms and finding the most probable $a$-$n_{\mathrm{p}}$ combination (Fig. 1F), with precisions of 20 nm and 0.002, respectively. The usefulness and precision of this self-calibration are highlighted through experiments on both polystyrene beads and oil droplets. The control experiments are conducted with polystyrene beads of nominal radii of $1.55\pm 0.03\mathrm{\mu }\mathrm{m}$ (13 experiments) and $2.965\pm 0.15\mathrm{\mu }\mathrm{m}$ (13 experiments). Measured radii are $1.50\pm 0.02\mathrm{\mu }\mathrm{m}$ and $2.98\pm 0.03\mathrm{\mu }\mathrm{m}$, respectively, where all uncertainties correspond to a 99-% CI. So, the nominal and measured values are in agreement up to experimental errors i.e. $20\mathrm{nm}$. As for $n_{\mathrm{p}}$, these experiments give $n_{\mathrm{p}}=1.589\pm 0.005$, again in agreement with the expected value. Now, over 32 distinct experiments on droplets of unknown size, one gets $n_{\mathrm{p}}=1.461\pm 0.003$ (average $\pm$ SD), which is in agreement with the nominal value up to 0.1 % (relative error).

Second, independently of the parameters that affect the optical Lorenz-Mie theory, other parameters can be measured in situ thanks to the available trajectories. Specifically, before getting confined, a droplet falls toward the wall. Its sedimentation speed $v_{\mathrm{s}}$, arising from the balance of viscous dissipation and gravity acceleration $g$, reads:

$$\begin{array}{c}\hfill v_{\mathrm{s}}=\frac{2}{9}g{a}^2\frac{\mathrm{\Delta }\rho }{{\eta }_{\mathrm{m}}}.\end{array}$$ [5]

Thus, measuring $v_{\mathrm{s}}$ gives the buoyant density $\mathrm{\Delta }\rho$, i.e. the mismatch between the medium and the falling droplet, as long as the liquid’s bulk viscosity ${\eta }_{\mathrm{m}}$ is known. Both $v_{\mathrm{s}}$ and ${\eta }_{\mathrm{m}}$ are measured in situ from the sedimentation. The former, $v_{\mathrm{s}}$, is computed through the vertical drift $⟨{\delta }_{\tau }z⟩$ (SI Appendix, Fig. S1A), where ${\delta }_{\tau }z=z(t+\tau )-z(t)$ denotes a vertical displacement at a time $t$ and over a time delay $\tau$, and $⟨·⟩$ the average over time. The latter, ${\eta }_{\mathrm{m}}$, is computed through the second-order moment $⟨{\delta }_{\tau }{x}^2⟩$ of the horizontal displacement ${\delta }_{\tau }x$ (SI Appendix, Fig. S1B). For a falling colloid, $⟨{\delta }_{\tau }z⟩=v_{\mathrm{s}}\tau$, and $⟨{\delta }_{\tau }{x}^2⟩=2D_0\tau$, where $D_0=k_{\mathrm{B}}T/(6\pi {\eta }_{\mathrm{m}}a)$ denotes the bulk diffusion coefficient and $k_{\mathrm{B}}T$ the thermal energy. Note that the cutoff of the full trajectory is chosen so that the colloid remains at a distance from the wall ten times greater than $a$, in order to ensure measuring bulk properties.

Finally, those in situ microscale measurements are compared to ex situ macroscale rheological ones. Each liquid is sheared in a cone-plate geometry of a rheometer (HR20 from TA Instruments©), 10 times, with shear rates between $100{\mathrm{s}}^{-1}$ and $1,000{\mathrm{s}}^{-1}$. The in situ and ex situ measurements, summarized in SI Appendix, Table S2, are consistent with each other up to 10 %, as are the resulting density mismatches $\mathrm{\Delta }\rho$ and the expected ones from literature (64, 65), up to the experimental error.

Soft-Lubrication Model.

The main points precising the scaling discussed in the main text for the soft Brownian force are given hereafter. Specifically, the deterministic expression for the lift force acting on an immersed droplet moving horizontally near a rigid wall is first derived, and then adapted to the fluctuating case through a simple argument on the velocity scale. The complete derivation is provided in SI Appendix.

We specifically focus on the case where the viscosity of the droplet, denoted by ${\eta }_{\mathrm{p}}$, is much larger than the viscosity of the liquid in the outer medium. Along with a no-slip criterion at the interface of the droplet, this corresponds to the droplet translating with the velocity of the center of mass of the droplet like a rigid bead. The velocity of the center of mass is denoted by $V_0$ and is aligned with the $x$-axis. To highlight the influence of the boundary on the droplet’s motion, we focus on the lubrication regime where the gap distance $z\ll a$. Given that the dimensions of the droplet are on the order of a few microns and that it translates with a small velocity of the order of the thermal velocity, the Reynolds number $Re={\rho }_{\mathrm{m}}{V}_{0}a/{\eta }_{\mathrm{m}}$ is much smaller than unity, allowing us to ignore inertial effects in the fluid.

The fluid flow is described by Stokes’ equations and the incompressibility condition, associated with no-slip boundary conditions at both the droplet surface and the rigid boundary. The interface of the droplet deforms based on the Young–Laplace’s law allowing us to define compliance $\kappa =\text{Ca}/{ϵ}^3$ of the interface deformation of the droplet, where $\text{Ca}={\eta }_{\mathrm{m}}{V}_0/\gamma$ is the capillary number which quantifies the strength of viscous dissipation in comparison to surface tension, and $ϵ=z/l$ denotes the standard lubrication parameter. For small compliance, the lubrication pressure can be calculated perturbatively with the leading order pressure field corresponding to that near a rigid boundary (66). Since the nature of the interface deformation can be calculated, the perturbative correction to the pressure field dependent on the interface deformation can then be calculated as well. This $O(\kappa )$ pressure field leads to the soft-lubrication vertical force $F_{+}$. After integration:

$$\begin{array}{c}\hfill F_{+}\simeq 16\pi \left(\frac{{\eta }_{\mathrm{m}}^2{V}_0^2{a}^3}{\gamma z^2}\right){\int }_0^{\infty }R{P}_{10}\text{d}R=\frac{6\pi }{25}\frac{{\eta }_{\mathrm{m}}^2{V}_0^2{a}^3}{\gamma z^2}.\end{array}$$ [6]

Using the squared thermal velocity $3k_{\mathrm{B}}T/(2\pi a^3{\rho }_{\mathrm{p}})$ (for two horizontal degrees of freedom) as an estimate of $V_0^2$ in the fluctuating case, plugging it into Eq. 6, and adding an ad hoc exponential cutoff factor to ensure long-term equilibrium, one finally gets a minimal Ansatz for the soft-Brownian force:

$$\begin{array}{c}\hfill F_{\mathrm{nc}}\approx 0.36\frac{{\eta }_{\mathrm{m}}^2{k}_{\mathrm{B}}T}{\gamma {\rho }_{\mathrm{p}}{a}^2}{\left(\frac{a}{z}\right)}^{2}exp\left(-\frac{\tau }{{\tau }_{\mathrm{eq}}}\right),\end{array}$$ [7]

where ${\tau }_{\mathrm{eq}}$ is the equilibration time.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

See SI Appendix for dataset descriptions. Data have been deposited in GitHub (https://github.com/EMetBrown-Lab/Brownian-motion-near-soft-interfaces) (58). Some study data are available (Movie S1 was trimmed from the one leading to Figs. 2, 3, and 4A of the main text, due to heavy size constraints and is shown here for illustration purposes. The 25-Go full video is available upon request to the authors due to size constraints, but the full trajectory is available (Dataset S1). The trimmed video is also available at https://github.com/EMetBrown-Lab/Brownian-motion-near-soft-interfaces).