Hydrodynamic shear dissipation and transmission in lipid bilayers.

Vital biological processes, such as trafficking, sensing, and motility, are facilitated by cellular lipid membranes, which interact mechanically with surrounding fluids. Such lipid membranes are only a few nanometers thick and composed of a liquid crystalline structure known as the lipid bilayer. Here, we introduce an active, noncontact, two-point microrheology technique combining multiple optical tweezers probes with planar freestanding lipid bilayers accessible on both sides. We use the method to quantify both fluid slip close to the bilayer surface and transmission of fluid flow across the structure, and we use numerical simulations to determine the monolayer viscosity and the intermonolayer friction. We find that these physical properties are highly dependent on the molecular structure of the lipids in the bilayer. We compare ordered-phase with liquid disordered-phase lipid bilayers, and we find the ordered-phase bilayers to be 10 to 100 times more viscous but with 100 times less intermonolayer friction. When a local shear is applied by the optical tweezers, the ultralow intermonolayer friction results in full slip of the two leaflets relative to each other and as a consequence, no shear transmission across the membrane. Our study sheds light on the physical principles governing the transfer of shear forces by and through lipid membranes, which underpin cell behavior and homeostasis.

Lipid molecules self-assembled into a bilayer structure form the matrix of cell membranes (1) and enclose the entire cell as well as membrane-bound organelles. In a bilayer, lipids are aligned together, free to diffuse laterally, and experience thermal fluctuations, thus exhibiting properties akin to that of an ordered liquid crystalline sheet (2). The fluidity of the membrane is paramount to understanding cellular dynamics, as it supports membrane protein diffusion and reorganization (1). Hydrodynamic shear stresses at the interface between lipid bilayers and their aqueous environment influence the biochemical responses underlying critical functions (3), such as dictating cell shape (4), sensing (5), adhesion and locomotion (6, 7), cell division (8), and transport of molecules and ions (9–12).

The shear rheology of lipid bilayers has been investigated using passive microrheology (13–17), active microrheology with surface shear rheology (18, 19), atomic force microscopy (20), magnetically (21, 22) and optically induced fluctuation analysis (23), nanotube pulling (24, 25), fluorescence lifetime imaging of membrane-bound molecular rotors (26), and magnetic (27) and optical tweezers (28). These measurements have made use of model lipid bilayer systems including supported bilayers (29), giant unilamellar vesicles (30), Langmuir troughs (31), nanodiscs (32), and freestanding (or black) bilayers (33, 34). While these methods have contributed significantly to our understanding of lipid membranes, they often trigger out-of-plane deformation modes (e.g., out-of-plane stretching and bending), in addition to in-plane shear deformation. Many of the passive probes used are embedded within the lipid bilayer and may alter the local physical and chemical properties (35). Furthermore, some model lipid bilayers used do not allow symmetric access to both sides of the bilayer. In such systems, the surface shear rheometry of lipid bilayers often measures the combined resistance to in-plane shear in each monolayer, characterized by the monolayer surface viscosity (36) and intermonolayer friction coefficient (22, 37), which represents the resistance to shear across the bilayer.

Here, we perform active, noncontact, two-point microrheology to characterize the viscous and frictional properties of the bilayer. Our approach uses a planar freestanding lipid bilayer (38, 39). The planar geometry of the freestanding membranes allows us to decouple in- and out-of-plane deformations, which facilitates the interpretation of the measurements using hydrodynamic models (22, 36, 37). Additionally, it enables the symmetric interfacing with double optical tweezers. Such an approach affords us the opportunity to probe both sides of the lipid bilayer symmetrically and to impose a local shear deformation both in the plane of and across the membrane. We perform two independent measurements: the drag force on a sphere translating parallel to the bilayer and the flow velocity induced on the other side of the bilayer. The first measurement allows us to quantify fluid slip in the vicinity of the membrane, while the second one characterizes the transmission of flow and shear on the other side of the bilayer. Combining these two measurements, we determine the monolayer viscosity and the intermonolayer friction coefficient from established models (37, 40). We vary the lipid bilayer composition and provide insights into the physical roles of viscosity and friction and how they relate to lipid composition and phase.

Results

Interfacing Lipid Bilayers with Optical Tweezers.

Freestanding lipid bilayers are formed in microfluidic devices following a previously described protocol (38, 39) (). The microfluidic devices consist of two parallel rectangular microchannels (100-m high and 500-m wide) with several apertures between the two channels, where lipid bilayers are formed (Fig. 1). The apertures extend over the entire 100-m height of the channel and are 85-m wide, but they decrease to 65 m due to the swelling of the polymer during the formation of the lipid bilayer (Fig. 1 ). The versatility of the flow device supports the formation of lipid bilayers with different compositions, extending over the entire surface of the aperture. Here, we study membranes with two homogeneous lipid compositions, one in an ordered phase using 1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC) and one in a liquid-disordered phase using 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) (41). In addition, we also form membranes with a mixture of DOPC/DPPC with a 2:1 M ratio. The flow cells are mounted on a piezostage and placed under an inverted microscope. The surface of the membrane is parallel to the optical axis of the microscope, such that the bilayers can be interfaced with optical tweezers. The tweezers are used to trap polystyrene microspheres of radius and m added to the aqueous solution. We use a double-optical trap setup using an Nd:YAG (neodymium-doped yttrium aluminum garnet; 1,064-nm wavelength) laser and measure forces on one trap using back focal plane interferometry (42). The double-trap setup supports the trapping of polystyrene microspheres on each side of the bilayer.

Fig. 1.

Interfacing lipid bilayers with optical tweezers. (A) Overview of the microfluidic device used for lipid bilayer formation and optical tweezers experiments. (B) Close-up view of one aperture within the microfluidic device with a lipid bilayer spanning across it. (C) High-magnification optical image of a lipid bilayer spanning across an aperture. (D) Schematic and (E) image of the experiment for measuring the drag force. (F) Schematic and (G) overlaid image sequence of the experiment for measuring flow transmission. (G) The red dashed arrow represents the direction of motion. The images were captured using a light microscope and overlaid with a time step of 22 ms between images. Yellow dashed lines are used at the edges of the apertures in E and G to help visualize the bilayer. (Scale bars: A, 5 mm; E and G, 10 m.)

Drag on a Sphere in the Vicinity of a Bilayer.

The shear boundary condition at the surface of a lipid bilayer has been represented using various assumptions. Such assumptions include treating the bilayer as a solid interface (43–45), an incompressible two-dimensional (2D) fluid (15, 16, 19, 36, 40), and two incompressible 2D fluids coupled by intermonolayer friction (14, 18, 22, 24, 25, 37). The nature of the boundary condition strongly influences the fluid flow in the vicinity of the bilayer. For example, the drag experienced by a translating sphere is significantly increased by the presence of a solid interface. To characterize the shear boundary condition on a bilayer, we use measurements of the drag force on a sphere. A polystyrene bead of known radius within a viscous fluid of viscosity is translated parallel to the lipid bilayer, at a prescribed velocity relative to the bilayer. In our experiments, rather than translating the bead with the optical tweezers, we equivalently displace the membrane by moving the microfluidic device with the piezostage while keeping the bead in place with the optical tweezers (Fig. 1 and Movie S1). We deduce the drag force on the bead by measuring the bead displacement from the focal point of the optical tweezers (46) (). Experiments are repeated at increasing distances from the interface, .

Fig. 2 reports our measurements of for a rigid glass interface (red Xs) and for bilayers of varying composition, including ordered DPPC membranes (black triangles), liquid-disordered DOPC membranes (blue circles), and DOPC/DPPC mixtures (green squares). Far from the bilayer, we find corresponding to the drag on a sphere in an unbounded viscous fluid. The increase in drag force at closer distances from the bilayer strongly depends on the nature of the interface. For the solid glass interface, we observe a sharp increase in drag as (49), reaching a value of close to the interface at . The drag increase close to a lipid bilayer is much more moderate and depends on the lipid composition (Fig. 2). We quantify the drag increase by the ratio between the measured drag, when the bead almost touches the bilayer for , and Stokes’ drag . We find values for the drag ratio coefficient ranging from for ordered DPPC membranes to for disordered DOPC membranes. For the DOPC/DPPC membranes, we find an intermediate value of .

Fig. 2.

Drag force and flow transmission for a sphere translating parallel to lipid bilayers. (A) Relationship between the coefficient of drag and the distance from the lipid bilayer and solid glass wall. The model is from Eq. (47). (B) Relationship between the flow speed at a distance from the moving sphere and the translation velocity of the sphere. The model is from Eq. (48). Error bars represent SDs from separate experiments performed on different membranes formed within different microfluidic devices.

The marked differences in drag force between the interfaces signal differences in the applicable shear boundary conditions. Interactions at a fluid–structure interface can be represented by a slip length (50), which is the distance required to extrapolate the velocity profile into a rigid wall in order to obtain the same stress condition. To quantify fluid slip near the various interfaces, we introduce to the prediction of drag force on a sphere near a solid interface when solving Stokes’ equations as follows:For a no-slip boundary condition, the model coincides with Faxén’s law (47) when , represented by the red dashed line in Fig. 2. Our drag measurements near the solid glass interface are in agreement with a no-slip boundary condition. By fitting Eq. to the measurements for each lipid type, we can determine their respective slip lengths . For the ordered-phase DPPC bilayers, the slip length , and the drag force is closest to that of the no-slip boundary. For the liquid disordered-phase DOPC bilayers, the slip length , and the drag force does not increase significantly near the bilayer. For the mixed (DOPC/DPPC) bilayers, the slip length , and the drag force lies between the values measured for DPPC and DOPC independently. The lipid bilayers exhibit fluid slip, as evidenced by the nonzero values of , and the membranes are not accurately represented by a solid interface with the no-slip boundary condition. Our results support previous observations on lipid bilayers, where these fluid-like structures exhibit viscous (36) and frictional resistance (37), in addition to the tensile and bending resistance expected for thin films (51).

Flow Transmission and Slip across a Bilayer.

We now characterize the transmission of shear stress from one side of the bilayer to the other and therefore, the interlayer interactions between the two leaflets of the bilayer. Quantifying this property requires experimental access to both sides of the bilayer. For these measurements, we use two optically trapped spheres, on each side of the bilayer (Fig. 1 ). We trap one sphere on one side of the bilayer and translate it, at a given velocity parallel to the interface, in order to generate a local fluid flow in the vicinity of the bilayer. The motion of the bead is induced by deflecting the laser beam with an acousto-optic deflector, which results in the translation of the trap center. We use a second, smaller sphere on the other side of the bilayer for velocimetry (46) to measure the flow field transmitted across the bilayer (Fig. 1 and Movie S2). The center to center distance between the two optical tweezers is m. For each lipid bilayer composition, we repeat the experiments several times using increasing values for the translation velocity m and record for each experiment the flow velocity measured on the other side of the membrane.

We first report the fluid velocity in the absence of a lipid bilayer, when the two beads are only separated by the aqueous phase. These measurements can be compared with Stokes’ flow around a translating sphere:Our measurements are in perfect agreement with the theoretical prediction from Eq. , represented by the red dashed line in Fig. 2. Strikingly, in the presence of lipid bilayers, the range of velocities reported covers the extremes of what could be expected. The black triangles and blue circles are our measurements when an ordered DPPC bilayer and a disordered DOPC bilayer, respectively, are between the translating and probing spheres. For the DPPC bilayer, we measure to be zero for the full range of velocity . These measurements correspond to a complete absence of flow transmission, such that the fluid on the other side of the bilayer remains stagnant. For the liquid-disordered DOPC bilayer, we measure a significant flow transmission, with the flow velocity reaching similar values as those measured when the beads are only separated by the aqueous phase and no lipid bilayer is present. The transmission of flow from one side of the bilayer to the other can be represented by the transmission coefficient , defined as the ratio between the velocity measured by the probe and the velocity predicted from the Stokes’ equation at the probe location. For the DPPC membranes, we find , while for the DOPC membranes, we find . For the DOPC/DPPC mixtures, we find partial transmission, with . Our results indicate that there is almost no slip between the monolayers of the homogeneous DOPC bilayers. Remarkably, these results suggest that there is full slip between the leaflets of the ordered DPPC bilayers and partial slip for the DOPC/DPPC mixtures.

Surface Shear Rheology.

We proceed by determining the surface shear rheology of lipid bilayers, characterized by two properties: the monolayer viscosity and the intermonolayer friction coefficient . We can deduce both of these properties from the two independent measurements of the drag ratio coefficient and the flow transmission coefficient . For this purpose, we model the flow in our flow cell and in the bilayer following the classical model of Saffman (40) widely applied to flow involving lipid bilayers (19, 52). The geometry of the freestanding lipid bilayer simplifies the problem, as out-of-plane membrane deformations do not need to be taken into account. Indeed, for our observations, we find that the ratio of membrane tension to viscous forces, characterized by the capillary number, is negligible, , with N previously measured for a DOPC/DPPC bilayer prepared using the same method (39). Therefore, we do not expect the bilayer to deform. As such, we do not observe significant deformation of the bilayer during the experiments and assume it to remain planar in our model.

The aqueous phases on each side of the membrane are modeled as two three-dimensional incompressible fluid domains of viscosity ; = (, , ); and = (, , ), satisfying the Stokes’ equations, and separated by a planar interface representing the lipid bilayer (Fig. 3). The membrane includes two planar monolayers, which we model each as a 2D incompressible fluid domain with a surface viscosity (40): and , each satisfying the 2D Stokes’ equations (Fig. 3). Here, represents the viscosity of the entire bilayer. The dissipative forces exerted on the membrane due to the bulk flow in the aqueous phase and due to intermonolayer friction appear as an additional interfacial stress in the 2D Stokes’ equations, with , where is the unit vector in the direction and is the bulk hydrodynamic stress tensor (22, 37, 52). Finally, we impose a no-slip and no-penetration boundary condition at each interface between the aqueous phase and lipid bilayer (i.e., , , and ). Flow is generated through the motion of a sphere, which is treated as a solid interface. Details, including the governing equations and all boundary conditions, are provided in .

Fig. 3.

Hydrodynamic model of lipid bilayers. (A) Schematic of the computational domain with a sphere of radius translating parallel to a lipid bilayer of thickness with a velocity at a distance away. The domain is split into upper aqueous , upper lipid monolayer , lower lipid monolayer , and lower aqueous . (B and C) Velocity profiles for the plane passing through the center of the sphere. (D and E) Velocity profiles for the planes coinciding with the upper and lower monolayers. The velocities and are normalized by the translating velocity of the sphere. The profiles are for (B and D) Pasm and Pas and for (C and E) Pasm and Pas.

We perform the numerical simulations for a range of membrane viscosities Pasm and friction Pas (). Fig. 3 represents typical flow fields obtained from our numerical simulations. Fig. 3 corresponds to conditions for which no flow is transmitted, while Fig. 3 corresponds to a full transmission of the flow. In Fig. 3, the absence of flow in the aqueous phase on the opposite side of the bilayer to the moving bead is due to a velocity discontinuity across the bilayer, characteristic of a slip between the leaflets. In Fig. 3, the velocity field is continuous across the membrane due to the near absence of slip between the leaflets and to full flow transmission. The slip between the monolayers can be visualized by representing the flow within each leaflet of the bilayer (Fig. 3 ).

Our numerical results are consistent with known flow regimes. We first consider the drag ratio coefficient . For large values of the membrane viscosity , the bilayer has an effect on the flow similar to that of a solid membrane, and the values for approach the value of 2.4 obtained for a no-slip condition (Fig. 4). Decreasing the membrane viscosity leads to a decrease in drag . For very small values of , the dissipation is mostly due to the bulk viscosity of the aqueous phase . In this case, a drag smaller than the Stokes’ drag is expected for small values of , consistent with the bead being close to a free slip boundary, and a drag value close to the Stokes’ drag is expected for large values of , when there is no slip between the two leaflets and the transmission of shear is similar to that of an equivalent aqueous layer. We consider the flow transmission next (Fig. 4). For small values of , there is full slip between the monolayers and . Increasing corresponds to an increase in . It is noteworthy that is also a function of since a larger membrane viscosity leads to more dissipation and therefore, to lower flow amplitudes on the other side of the bilayer. For large values of , there is no slip between the monolayers, and the transmission decreases from one, when the membrane viscosity is negligible, to zero, when the bilayer is viscous and acts as a solid membrane.

Fig. 4.

Numerical simulations of drag force and flow transmission for a sphere moving parallel to lipid bilayers. Contour plots representing how (A) coefficient of drag and (B) coefficient of transmission (where is given by Eq. ) vary with monolayer viscosity and interlayer friction . (C) Isocontours of the coefficient of drag (dashed lines) and the coefficient of transmission (solid lines). The shaded regions represent the experimental values (averages and SDs from Fig. 2 and slopes and 95% CIs from Fig. 2) for DPPC (black), DOPC (blue), and DOPC/DPPC (green).

Isocontours of and intersect for a wide range of values, and the determination of and requires both measurements of . The method is most accurate for Pas and Pasm. For example, we find for DOPC/DPPC mixtures, Pasm and Pas at the intersection between the isocontours and (Fig. 4). The uncertainty on the measurements of is associated with a range in the determination of , which is represented as the shaded areas in Fig. 4. For pure DPPC, we find Pasm and Pas, and for pure DOPC, Pasm and Pas. For the DOPC/DPPC mixtures, Pasm and Pas.

Discussion

The bilayer viscosities measured with our method are in agreement with values from previous studies (14, 15, 19, 53–58). In particular, we report a viscosity of Pasm for ordered-phase DPPC membranes and Pasm for DOPC/DPPC mixtures, in agreement with a previous study of flow patterns inside vesicles reporting Pasm for liquid-ordered vesicles and Pasm for a comparable DOPC/DPPC mixture (19). The viscosity measured for pure DPPC membranes further agrees with measurements on DPPC liposomes obtained from the fluorescence lifetime imaging of membrane-bound molecular rotors, although our value is slightly higher (26, 59). Other studies of different mixtures have reported viscosities ranging from Pasm (53, 54, 56–58, 60), in agreement with the viscosity we report for DOPC/DPPC mixtures. Finally, for pure DOPC bilayers, the range we deduce for the viscosity Pasm is consistent with the lower viscosities previously reported for DOPC membranes Pasm (14, 55). It is noteworthy that for the mixture, we measure an intermediate viscosity between the values measured for the two homogeneous membranes, as has been found with other two-point microrheology approaches (60).

Fewer studies have attempted to measure the intermonolayer friction coefficients and have yielded values of Pas (14, 25, 61, 62). For pure DOPC, we report full transmission of shear and close to no slip between monolayers, corresponding to a regime for which the extent of shear transmission is comparable with that of an equivalent layer of the bulk aqueous phase. In this limit, isocontours of and are almost parallel (Fig. 4), and the accuracy of our method is limited. We can accurately report a lower bound Pas, and we note that larger intermonolayer friction, on the order of Pas, would correspond to values of and within the error bars of our measurements and is therefore consistent with our results. For the DOPC/DPPC mixtures and the DPPC bilayers, however, we find in both cases , which is many orders of magnitude lower than previously reported for other lipid compositions of 1-palmitoyl-2-oleoyl-glycero-3-phosphocholine (POPC) and 1-stearoyl-2-oleoyl-sn-glycero-3-phosphocholine (SOPC) membranes (61, 62). While our values are surprisingly low, they fall within the range where our two-point active microrheology method is the most accurate, in particular for the DOPC/DPPC mixtures. If, for example, we take as a premise the range of membrane viscosity previously reported for mixtures Pasm, then our separate measurements of and for DOPC/DPPC membranes provide two independent estimations for , both being consistent and smaller than Pas. In our study, the analysis required to fit and is greatly simplified by the planar geometry of the freestanding bilayer and does not require any assumptions on the values for the material properties of the membrane. We consider the membrane to behave as a 2D Newtonian fluid and assume a simple linear response for the intermonolayer stresses characterized by . In previous work on freestanding bilayers, the intermonolayer friction has been estimated from the rapid relaxation times s of shape fluctuations of wavelength smaller than 2.5 nm (61, 62). In contrast, our experimental measurements involve slower dynamics on the order of s and planar geometries. It is conceivable that the interactions and overlap between the hydrocarbon chains of the two leaflets will affect the membrane dynamics differently when considering fluctuations on a length scale smaller than the size of the lipids themselves, compared with the slow shearing of a planar bilayer. The low viscosities that we measure compared with refs. 61 and 62 are a possible indication of a rate- and curvature-dependent rheology for the friction between the monolayers, with a thickening behavior at higher rates and larger deformations. Within the shear rates that we could reach by varying , we do not observe such thickening behavior, although we note a moderate increase of for the measurement at the highest shearing rate (Fig. 2).

Conclusion

We have developed a method to characterize the shear rheology of lipid bilayers. By measuring how a lipid bilayer both dissipates () and transmits () hydrodynamic shear, we quantified the rheological properties of the membranes, the bilayer viscosity , and intermonolayer friction . The use of planar freestanding bilayers simplifies the analysis and modeling, such that no assumptions are required regarding the out-of-plane deformation and dynamics of the bilayer. We report measurements for three different lipid compositions in different phases and find values of membrane viscosities in agreement with previous studies. The same experiments yield extremely low values for the friction coefficient , which indicate a significant slip between the two monolayers when an external shear flow is applied. These low values contrast with the higher ones obtained in previous studies and bespeak a rate-dependent friction rheology more complex and richer than previously assumed.

Our results highlight how the viscous and frictional properties of lipid bilayers significantly affect the transition of shear stress through the bilayer and the drag force experienced by small [ O(m)] objects in the near field. Our hydrodynamic model and experimental platform may be extended to incorporate more complexity, such as lipid mixtures, asymmetric bilayers, and embedding of other macromolecules, in order to more closely approach the in vivo conditions of cells, as well as to investigate possible nonlinear rheological behaviors of lipid bilayers.

Materials and Methods

Microfluidic Channel Fabrication.

Microfluidic devices were fabricated following the technique from Marin and coworkers (38, 39). Specifically, we prepared devices with two parallel rectangular microchannels (100-m high and 500-m wide) with several 85-m-wide apertures where the two channels meet and lipid bilayers can form (Fig. 1). We cast the devices in Norland Optical Adhesive 81 (NOA81; Norland Products) from polydimethylsiloxane molds. The NOA81 channels were sandwiched between a microscope glass slide (Corning Incorporated; 75 × 25 mm) and a glass coverslip (Menzel-Gläzer; size: 60 × 24 mm, thickness: m, no. 1.5). After being sealed, the microfluidic channels were functionalized using trichloro(1-perfluorooctyl) silane (Sigma-Aldrich) at 1.5% (vol/vol) in isooctane.

Lipid Bilayer Formation.

Two types of lipids were used to form the lipid bilayers, DPPC and DOPC, purchased from Avanti Lipids. For the pure compositions, the lipids were used in their stock concentrations: 10.0 mg in chloroform. For the DOPC/DPPC mixture, they were mixed to a 2:1 M ratio at a 37.5-mg total lipid concentration in chloroform. The bilayers were formed by first flowing the organic (chloroform) phase containing the lipids, followed by an aqueous phase. The aqueous phase consisted of 10 mM 4-(2-hydroxyethyl)-1-piperazineethanesulfonic acid (HEPES; titrated to pH 7.4 using 150 mM KCl) mixed with 0.5 mg bovine serum albumin and polystyrene microspheres of radius and m (Polysciences Inc.) diluted to 1:100,000 and 1:25,000, respectively. The fluids were pushed through the microchannels using a syringe pump (ProSense NE-300) with flow rates between L . This technique has been shown to reproducibly form lipid bilayers that can be interfaced with optical tweezers (39). The experiments were conducted at room temperature (21 C), where the pure lipids have been reported to be in liquid-ordered phase (DPPC) and liquid-disordered phase (DOPC) (41).

Optical Tweezers Experiments.

We used optical tweezers to both measure the drag force on a particle translating near a lipid bilayer and the flow it generates on the other side of the bilayer. The methodology follows from previous studies measuring the surface tension of lipid bilayers (39) and the flow fields generated by swimming cells (63, 64). The optical tweezers used in the study are similar to those in ref. 42. Namely, an Nd:YAG laser (IPG YLR-10-LP-Y12; 1,064-nm wavelength, 10-W maximum power) was focused through a water immersion objective (Nikon CFI Plan Apo VC 60× numerical aperture [NA] = 1.20) to generate a trapping force , where is the bead displacement from the center of the trapping laser and is the trap stiffness. The bead displacement was measured at a sampling frequency of 50 kHz using back focal plane interferometry with a detection laser (TOPAG LDT-830-30 GC; 830-nm wavelength). The optical traps had stiffness values of pN . The trapped bead experiences at most 300 mW of irradiation from the lasers that may result in 3 C of local heating (65). For a temperature difference of 3 C, the timescale for heat transfer through diffusion is much lower than that of convection; therefore, it is insufficient to trigger convective flows.

Since the bead Reynolds number is low and inertia, added mass, and Basset forces are negligible (66), the trapping force and hydrodynamic drag are balanced in steady state. Therefore, by measuring the trapping force, we determined the drag acting on the bead in the direction of motion, or . Similarly, the velocity at the location of the second trapped bead (Fig. 1) was determined from the measured bead displacement . Through a force balance between the trapping laser and hydrodynamic drag, the flow velocity in can be expressed as .