Time dependence of advection-diffusion coupling for nanoparticle ensembles
Alexandre Vilquin, Vincent Bertin, Pierre Soulard, Gabriel Guyard, Elie Raphaël, Frederic Restagno, Thomas Salez, Joshua Mcgraw
TTime dependence of advection-diffusion coupling for nanoparticle ensembles
Alexandre Vilquin, Vincent Bertin,
1, 2
Pierre Soulard, Gabriel Guyard,
1, 3
Elie Rapha¨el, Fr´ed´eric Restagno, Thomas Salez,
2, 4 and Joshua D. McGraw Gulliver CNRS UMR 7083, PSL Research University,ESPCI Paris, 10 rue Vauquelin, 75005 Paris, France Univ. Bordeaux, CNRS, LOMA, UMR 5798, F-33405, Talence, France Universit Paris-Saclay, CNRS, Laboratoire de physique des solides, Orsay, France Global Station for Soft Matter, Global Institution for CollaborativeResearch and Education, Hokkaido University, Sapporo, Hokkaido, Japan (Dated: July 17, 2020)Particle transport in fluids at micro- and nano-scales is important in many domains. As comparedto the quiescent case, the time evolution of particle dispersion is enhanced by coupling: i ) advectionalong the flow; and ii ) diffusion along the associated velocity gradients. While there is a well-known,long-time limit for this advection-diffusion enhancement, understanding the short-time limit andcorresponding crossover between these two asymptotic limits is less mature. We use evanescent-wavevideo microscopy for its spatio-temporal resolution. Specifically, we observe a near-surface zone ofwhere the velocity gradients, and thus dispersion, are the largest within a simple microfluidic channel.Supported by a theoretical model and simulations based on overdamped Langevin dynamics, ourexperiments reveal the crossover of this so-called Taylor dispersion from short to long time scales.Studying a range of particle size, viscosity and applied pressure, we show that the initial spatialdistribution of particles can strongly modify observed master curves for short-time dispersion andits crossover into the long-time regime. PACS numbers:
Hydrodynamic flows typically exhibit a spatially vary-ing velocity, often as a result of a nearby solid, immobileboundary. When microscopic particles are transportedby such a near-surface flow, the coupling between diffu-sion along the flow gradients and streamwise advectionleads to an enhanced dispersion as compared to the no-flow case. This enhancement was first quantitatively de-scribed by G.I. Taylor [1] for laminar flows in a cylindricaltube. These predictions were particularly applicable totimes long compared to the one over which a particle dif-fuses across the width of the tube, here called the Taylortime. From this foundational work, and in a strict anal-ogy to simple Fickian diffusion, a dispersion coefficientcould be identified as half the variance of the solute dis-placement divided by the time; the latter being quadraticin the typical flow speed. Taylor’s description was for-malised in this laminar tube-flow context by Aris [2], gen-eralised by Brenner and others [3–6], and eventually usedto measure diffusion coefficients [7] and influences lab-on-chip design [8].Such a mature theoretical description of the disper-sion complemented with experiments is not available fortimes short compared to the Taylor time. Neverthe-less, several theoretical works [9–13], and some morerecent experiments [14–16] were devoted to this short-time limit. These results demonstrate that the dis-persion coefficient quadratically increases with time forsingle-particle observations. However, it is often the casethat an ensemble of particles is released from a partic-ular region and their spatio-temporal evolution shouldbe known in detail. Drug delivery from a suddenly rup- tured nanoparticle [17, 18] provides just one microscopicexample. More generally, the efficacy of nanoconfinedchemical reactions [19, 20] should critically depend onthe time-dependent spatial distribution of reactants. Be-sides, Taylor dispersion also plays an important role inpulmonary air exchanges [21, 22], the mixing of cere-brospinal fluids [23], DNA molecular mobility [24] andother contexts at the micro- and nano-scales [25–28].Among the experiments capable of quantitatively ac-cessing short- and long-time dispersion at the nanoscaleis total internal reflection fluorescence microscopy(TIRFM). Initially developed for near-surface cell biol-ogy [29, 30], TIRFM illuminates a sample with an evanes-cent wave decaying exponentially from the surface asshown schematically in Fig. 1(a). TIRFM was partic-ularly used to study near-surface colloid/surface interac-tions in equilibrium [31], following which nanovelocime-try was implemented [32–36], allowing for the quantita-tive characterization of slip boundary conditions at thenanoscale [32, 37–39], and to study hindered diffusionnear a solid/liquid interface [40].Here we use TIRFM to examine the dispersion ofnanoparticle ensembles for a broad range of timescales ina linear shear flow, as shown schematically in Figs. 1(a)and (b). We first demonstrate the quadratic shear-ratedependence of the dispersion coefficient for a linear flowprofile. We then study the time dependence, wherefor short-times the dispersion coefficient increases withtime before reaching a long-time plateau predicted usingthe Taylor method. We find that the transient regimeis strongly affected by the grouping of particles: i ) in a r X i v : . [ c ond - m a t . s o f t ] J u l the most general case a mixed-power time dependenceis observed for the dispersion coefficient; ii ) a corre-sponding linear time dependence can be observed forall-channel, simple-shear protocols; and iii ) the pure,quadratic time dependence of the dispersion coefficientcan be approached by selecting particles in the finestrange of altitudes available to the experiment. Analyticalmodelling allows us to capture the asymptotic behaviours—including an early-time linear-to-quadratic crossover—while Langevin simulations are used to quantitatively re-cover the full experimental dynamics.In our experiments —the details of which are foundin the Materials and Methods, with the SupportingInformation (SI) containing all notations— fluorescentnanoparticles with radius a = 55 or 100 nm were ad-vected along the x -direction ( cf. Fig. 1(b) and SI Video 1)of a pressure-driven flow. Pressure drops 5 ≤ ∆ P ≤ h = 18 µ m, width w = 180 µ m, and length, ‘ = 8 . ≤ η ≤ . z = Π ln( I /I ) whereΠ is the exponential decay length, I is the measured par-ticle intensity, as shown in Fig. 1(b), and I is the fluo-rescence intensity at the wall. Optical aberrations leadto small deviations from an exact exponential decay forthe fluorescence intensity, and such details are discussedin the SI. Comparing subsequent images, sample trajec-tories as in Fig. 1(b) and SI Video 2 were constructedusing home-built Matlab routines.The experimental setup allows the observation of par-ticles in a range of altitudes a . z . µ m from thesolid/liquid interface, the solid being a glass coverslip.Here z = 0 is set at the solid/liquid boundary and thecamera sensitivity determines the upper z -limit. In prac-tice, we do not observe particles for z .
200 nm as aresult of electrostatic and steric interactions [31, 41, 42](see SI for details). The near-surface flow has the ad-vantage of simultaneously exhibiting the lowest velocities(0 − µ m s − ) and largest shear rates (100 −
600 s − )as compared to the rest of the channel, offering pertinentconditions for studying the advection-diffusion coupling.In order to obtain mean velocity profiles along theflow direction over a given lag time, τ , displacements,∆ x ( z, τ ) = x ( z, t + τ ) − x ( z, t ), were measured for eachpair of frames for which an identical particle was de-tected, and taking t as the initial observation time (seeFig. 1(b)). Since the particle intensity encodes the al-titude, we first sort the particles into a series of inten-sity bins, each bin corresponding to a range of approxi-mately 15 nm. Then, the lag-time-independent (verified)streamwise mean velocity v x (¯ z ) = h ∆ x (¯ z, τ ) i /τ is deter-mined. Here, h·i denotes ensemble averaging over ca. particle observations for each experimental condition ac-cessed for all t . The notation ¯ · denotes averaging over Figure 1: (a) Schematic of two Brownian colloids transportedby a near surface shear flow in a microchannel illuminated byan evanescent wave. (b) Superposition of TIRFM images withlag time τ = 12 . v x = h ∆ x i /τ for 55 nm-radius particlesas observed in SI Videos 1 and 2 with a lag time τ = 2 . γ . Inset: Shear rateas a function of pressure drop across the microfluidic channel.The dashed black line is a linear regression. all frames during the lag time.Figure 1(c) shows the streamwise velocity profiles for55 nm-radius particles in water and several pressuredrops. The solid lines show that the profiles are well ap-proximated by linear functions (see SI and [35, 43] for adiscussion of the small non-linearities). The spread of v x -intercept values arises from the spread of values for I ateach pressure, z = 0 being taken as the mean of Π log I over the different ∆ P ; z = 0 is thus resolved to within20 nm of the solid/liquid interface assuming no slip [35]as justified in the SI. The low Reynolds numbers (Re = ρhU/η ≈ − with ρ the fluid density and U the aver-age velocity in the whole channel) indicate a viscosity-dominated flow for which v x (¯ z ) = ∆ P (cid:0) ¯ z − h ¯ z (cid:1) / ηL , i.e. a Poiseuille flow. In the region z . µ m, and Figure 2: Probability density functions (PDFs) for all trans-verse (a) and streamwise (b) displacements measured for sev-eral pressure drops; the colour code corresponds to the one inFig. 1(c). The black line in (a) indicates a Gaussian model P (∆ y ) = exp (cid:0) − ∆ y / σ y (cid:1) / q πσ y . In (c) the ∆ y -PDFfor the largest pressure drop is decomposed into PDFs for sev-eral z . Each green-black PDF corresponds to a z -range of ca.
30 nm (some curves omitted for clarity), and the mean valueof z is indicated by the colour scale of (d). The decomposed∆ x -PDFs for the smallest pressure drop (thinnest green-blacklines) and the largest pressure drop (thickest green-blacklines) are shown in (d). All the displacements are measuredfor a time lag τ = 2 . given the channel height h = 18 µ m, the deviation of thePoiseuille profile from linearity is expected to be less than5 %. Therefore, at first order in ¯ z/h , we have v x (¯ z ) ’ ˙ γ ¯ z ,with the shear rate ˙ γ = ∂ ¯ z v x | = h | ∆ P | / ηL and ∂ ¯ z denoting partial differentiation with respect to ¯ z . Theinset in Fig. 1(c) shows the shear rate values extractedfrom the velocity profiles for several pressure drops. Ashighlighted by the dashed black line, the shear rate in-creases linearly with the pressure drop. The slope, givenby h/ ηL , provides a water viscosity η = 0 . ± . ∂ | ∆ P | ˙ γ is in quantitative agreementwith bulk rheological measurements of the viscosity forall of the liquids investigated here.Having discussed the mean velocity of the particles,we turn our attention to the displacement distributions.Figures 2(a) and (b) show the probability density func-tions (PDFs, here called P ) of the transverse (∆ y ) andstreamwise (∆ x ) displacements for 100 nm-radius par-ticles in water over a duration τ = 2 . z = z ( t ), the altitude at the initial ob-servation time. This z ( t ) should be distinguished fromthe average ¯ z over the lag time τ . In Fig. 2(a) it is shownthat the transverse displacement PDFs do not depend onthe pressure drop and are well described by a Gaussianover two decades. The global standard deviation pro-vides an approximation for the unidimensional Brownian diffusion coefficient σ y / τ ≈ . ± . µ m s − . De-spite the lubrication effect discussed below, this estimateis close to the value predicted by the Stokes-Einstein re-lation D = k B T / πηa ≈ . ± . µ m s − [45] forparticles with a = 100 nm advected in water, where k B is Boltzmann’s constant and T is the temperature and η was taken from bulk rheology. Contrasting with thetransverse displacement PDFs, those for the streamwisedirection in Fig. 2(b) are not Gaussian, become broad-ened with the pressure drop, and exhibit asymmetry asseen in Ref. [33].The TIRFM setup provides the particle distance fromthe glass/liquid interface through the detected intensity,allowing to distinguish the contributions of particles atdifferent altitudes to the global PDFs. In the transversedirection, the local PDFs are shown for the largest pres-sure drop in Fig. 2(c) and they are all Gaussian regardlessof z . In Fig. 2(d) are shown similar decompositions forthe smallest and largest pressure drops for streamwisedisplacements. These decompositions demonstrate thatthe asymmetry of the global distributions is mainly dueto the superposition of different mean displacements atdifferent altitudes. For the smallest pressure drop (yel-low) the local PFDs are only slightly shifted with in-creasing z due to the relatively low mean velocities, cf. Fig. 1(c). For the largest pressure drop (red), the meanvalues are shifted more strongly with increasing z as aresult of the higher shear rate. More importantly, the lo-cal PDFs thus provide access to the transverse diffusionalong y and streamwise dispersion along x for differentaltitudes.A detailed study of the local Taylor dispersion as func-tion of time and the various physical parameters at stakeis now described. In Figs. 3(a) and (b) are shown the lo-cal transverse diffusion coefficient, D y ( z, τ ) = σ y ( z ) / τ ,and the streamwise dispersion coefficient, D x ( z, τ ) = σ x ( z ) / τ . The latter comprises pure diffusive and ad-vection effects, and is remarkably larger (up to an order ofmagnitude) than the former. These data were obtainedfrom altitude decompositions as in Fig. 2 for several τ (red to blue). Figures 3(a) and (b) show that there isa general increase with z of D y and D x until reaching aplateau at large z ; in the SI, we show that these plateauvalues are in quantitative agreement with the Stokes-Einstein relation for all liquids investigated after com-paring to independently measured bulk viscosities. Thevariation with z in both directions is due to hydrody-namic interactions between particles and the solid/liquidinterface, leading to a hindered diffusion as discussed indetail elsewhere [40, 46–48]. The z -dependence of D y isin agreement with the prediction resulting from the ef-fective viscosity near a flat, rigid wall [46] (see the plainblack line in Fig. 3(a)). As expected, the transverse dif-fusion is not dependent on the lag time τ ; in contrast,the dispersion coefficients increase significantly with τ as Figure 3: (a) Transverse diffusion coefficients, D y , and(b) streamwise dispersion coefficient, D x , as a func-tion the apparent altitude and lag time, τ , for 100nm-radius particles with a pressure drop of 50 mbaracross the microfluidic device. In (a), the plain blackline corresponds to the the theoretical prediction D y = D (cid:0) − (9 / Z − + (3 / Z − − (45 / Z − − (1 / Z − (cid:1) with Z = z/a [46]. In (b) the dashed lines indicate the stream-wise dispersion coefficient, D h / i shown in (c) as a functionof lag time. From yellow to dark green, the bulk diffusioncoefficient increases. In particular —from bottom to top—the shear rates, particle radii and viscosities are { ˙ γ, a, η } = { , , . } , { , , . } , { , , } , { , , } inunits { s − , nm, mPa s } ; giving D = { } µ m s − determined from the Stokes-Einstein relation.(d) Reduced late-time dispersion coefficient versus thesquared shear rate; the solid lines have log-log slope 2. shown in Fig. 3(b).In Fig. 3(c) we show the lag-time dependence of thestreamwise dispersion coefficient. To do so, we define D h / i ( τ ) as the averaged D x ( z ( t ) , τ ) for all z ( t ) > H/ H , is the size of the observationzone. That is, particles beginning their trajectory in thetop half of the observation zone, thus limiting the afore-mentioned lubrication effects. In this figure, the bulkdiffusion coefficient D was varied by changing the par-ticle size and the liquid viscosity. For the lowest val-ues of D , D h / i continuously increases with time andthe temporal slope increases; by contrast, D h / i sat-urates to a plateau for the largest D . As explained by Taylor [1], the time needed to reach the dispersionplateau corresponds roughly to the time needed to dif-fuse across the channel height. Here the Taylor time istaken as τ z = H /D . In a rectangular channel, the exactcalculation (see SI) gives a characteristic diffusion time τ z /π . For the 55 nm-radius particles in water, assuminga length scale H ≈
700 nm, τ z /π ≈
13 ms, in reasonableagreement with the corresponding data of Fig. 3(c), with D = 3 . ± . µ m s − . For smaller values of D , thedispersion remains mainly in the short-time, increasing-slope regime. Nevertheless, taking the longest-time dataavailable (the data of Fig. 3(c) at τ = 50 ms, denoted D τ max ) for each D value, we next examine the shear-rate dependence of the dispersion.In Fig. 3(d) is shown the dependence of the reducedlate-time dispersion coefficient, D τ max /D −
1, for four D as a function of the shear rate. The solid lines (withslope 2 in log-log representation) show that the reduced D τ max increases quadratically with the shear rate ˙ γ forall D studied. To understand this result, we applied theclassical Taylor analysis ( i.e. long-time limit, see SI) toa linear shear flow in a rectangular channel, giving D x = D (cid:18) γ H D (cid:19) τ (cid:29) τ z , (1)for the infinite-time dispersion coefficient. Identifying˙ γH/ ≤ z ≤ H , Eq. 1 is equivalent to the usual resultof Taylor, i.e. D x = D (1 + α Pe ) [1]. In this latter ex-pression, the P´eclet number is Pe = U ( H/ /D , α is ageometry-dependent prefactor and H/ H ≈
500 nm,consistent with the range of z observed in Fig. 1(c). Be-cause the larger- D data does not reach the infinite-timeplateau, the prefactor for the linear regressions does notreveal the corresponding size of the flow region H . How-ever, as we show in the following, the quadratic shear ratedependence is preserved for all time regimes, explainingthe scaling of the data in Fig. 3(d).We now examine the detailed time dependence of dis-persion coefficient for all of the experimentally accessedtimes. Partly inspired by the shear-rate dependence ofFig. 3(d), we show in Fig. 4(a) the reduced D h / i nor-malised by ( τ z ˙ γ ) as a function of the dimensionless lagtime, τ /τ z . Remarkably, the data in Fig. 3(c), alongwith that for experiments implementing four other shearrates per D , collapse onto a single master curve. Sucha collapse suggests the existence of a universal functiondescribing the reduced dispersion coefficient. While inFigs. 3(c),(d) and Fig. 4(a), we consider particles begin-ning their trajectories in the top half of the channel, thisfraction can be generalised, with n representing the frac- Figure 4: Reduced dispersion coefficient as a function ofdimensionless time for all shear rates and D studied for frac-tions (a) n = 1 /
2, (b) n = 1, and (c) n → D , for thedata from dark green to yellow points are identical to thosein Figs. 3(c) and (d). The black dashed and solid lines re-spectively correspond to the asymptotic behaviours for thelong- and short-time regimes predicted by Eqs. 1 and 3. Thegray lines decorated with circles and triangles correspond tothe results from Langevin simulations for the n = 1 / n = 1and n = 0 conditions. For each panel, the correspondingsimulation results are highlighted in lighter gray. The insetsschematically show three instants of particle trajectories ad-vected from the associated observation zone in a linear shearflow and with diffusion in z . The slope triangles in (a) denotepower-law exponents 1 and 2. tion of the observation zone from which particles leave,with 0 ≤ n ≤
1. The reduced dispersion is thus expectedto follow a relation of the form (cid:18) D h n i D − (cid:19) ( ˙ γτ z ) − = F h n i (cid:18) ττ z (cid:19) . (2)Examining Fig. 4(a), we note that the reduced D h / i increases with time and then reaches a plateau. Accord-ing to Eq. 1, in the τ /τ z → ∞ limit, F h n i is expectedto reach 1 / τ /τ z ≈ τ dependence. A key foundation of this τ dependenceis the assumption that each particle begins its trajectoryat the same initial altitude. In general, however, parti-cles may leave from a non-peaked distribution of initialaltitudes. This distribution is particularly relevant forFig. 4(a), since the plotted quantity is related to an aver- age of all the particles in the top half of the observationzone (indicated by the dashed lines in Fig. 3(b)).To investigate the effect of such a distribution of initialparticle altitudes, we proceed by assuming that particlesare ‘released’ at time t and position z from a generaldistribution P ( z ) in an infinitely extended shear flow.This description should thus be valid for early times only.Given such a distribution of initial altitudes, the linearshear flow, and diffusion along the z -direction, we ana-lytically demonstrate that Eq. 2 is the appropriate formfor the lag-time-dependent, reduced short-time disper-sion coefficient (see SI). In the simple case for which par-ticles are uniformly released over a fraction n of the chan-nel height, the function F h n i becomes: F h n i (cid:18) ττ z (cid:19) = n (cid:18) ττ z (cid:19) + 13 (cid:18) ττ z (cid:19) , τ (cid:28) τ z . (3)The more general case of a non-uniform initial distribu-tion of altitudes is also captured by our model. In thatcase, the prefactor of the linear term in τ /τ z on the right-hand-side of Eq. 3 is replaced with the initial variance ofthe distribution normalized by the mean-square displace-ment in the vertical direction over the Taylor time: n → h z i − h z i D τ z . (4)We now examine the limits of Eqs. 3 and 4. A nullvariance of the initial distribution arises if all particlesstart at the same altitude (here called the “dot” condi-tion). For this dot condition, the classical τ dependencefor the reduced short-time dispersion coefficient is recov-ered, reflecting a steadily increasing diversity of newlysampled velocities (see SI Videos 3 and 4). The dot con-dition corresponds to single-particle tracking in a linearshear flow, as considered before [9–16]. For non-vanishinginitial variance, the reduced dispersion coefficient hasa linear temporal evolution at times shorter than thecrossover time τ C = 3( h z i − h z i ) / D obtained by set-ting the linear and quadratic terms of Eq. 3 to be equal.This linear behaviour for extended distributions resultsfrom particles at different altitudes transported differentdistances by the linear shear flow (SI Videos 3 and 4).When the initial variance is small with respect to H ,a crossover into the quadratic lag-time dependence oc-curs, before the Taylor plateau for the observation zoneis reached.In Fig. 4(a), we consider particles leaving from the up-per half of the observation zone, as schematically shownin the inset. The corresponding analytical prediction ofEqs. 2 and 3 with n = 1 / n = 1 / n = 0 (dot)and n = 1 (called the “line” condition), which at earlytimes are well outside the experimental data limits. Fur-thermore, we observe that the half-line simulation is par-allel to the line-condition simulation at the earliest times,and joins the simulation results for the dot and line con-ditions at later times. These observations are consistentwith the main features of the analytical model describedabove ( i.e. the linear-to-quadratic crossover). While theanalytical model and the simulation ignore electrostaticand hydrodynamic interactions with the wall, in the SIwe show that these phenomena respectively reduce thesize of the effective channel but do not affect the mainfeatures of time-dependent advection-diffusion couplingdiscussed here.In order to test our analytical model, we again lever-age the depth resolution of the TIRFM method to selectdifferent initial distributions. First, we choose the linecondition illustrated in the inset of Fig. 4(b). We thusstudy the dispersion coefficients for all observed parti-cles, meaning that we consider the global distributions ofFig. 2(b). The reduced dispersions measured for this con-dition are shown in Fig. 4(b). The results are once againin quantitative agreement with the early-time, analyticalprediction and the full-time Langevin simulations. Simi-larly, we consider particles leaving from a narrow altituderange, approaching the dot condition (inset of Fig. 4(c)).In Fig. 4(c) is shown the corresponding temporal evolu-tion of the reduced dispersion coefficient for z = 600 ± τ asymptotic behaviour predictedby Eq. 3 for n = 0. The experimental data also agreeswell with the corresponding Langevin simulation results.The data analysis and numerical simulations carried outfor three different initial particle distributions using thesame measurement data clearly demonstrate the crucialrole of such a distribution for the short-time dispersion.Furthermore, these results validate our novel analyticalmodel, that can be used to quantitatively rationalise gen-eral short-time dispersion observations.To conclude, we report on an experimental, theoreti- cal, and numerical study of advection-enhanced disper-sion from short to long times compared to the classicalTaylor time. We provide a quantative description of theshort- and long-time dispersion behaviours. First, weshow that the two regimes share the same shear-rate de-pendence, the shear rate being particularly large for near-surface transport where the advection-diffusion couplingshould be the largest. Furthermore, we reveal and char-acterize how the initial particle distribution affects theshort-time dispersion. Specifically, we observe a short-time, mixed-power-law behaviour for the general case,before a crossover to the well-known long-time satura-tion regime for linear shear flows. In the extremal casesof i ) full-channel observations, a linear approach of thedispersion coefficient to the long-time value is observed,while ii ) for fine depth resolutions a quadratic tendencyis approached. Altogether, the experimental data are inquantitative agreement with the analytical predictionsand results from Langevin numerical simulations. In therich context of particle transport, such concepts shouldprove pertinent in quantitative prediction and observa-tion of time-dependent, near-surface nanoparticle and so-lute dispersion, with applications related to microscopicbiology and nanoscale technologies. METHODS
All the experiments performed here employed pressure-driven flows (Fluigent MFCS-4C pressure controller) inmicrochannels with a rectangular section (height h = 18 µ m, width w = 180 µ m, length ‘ = 8 . µ m thick-ness constituting the bottom surface. The liquids usedwere ultra-pure water (18.2 MΩ cm, MilliQ) and water-glycerol mixtures that gave Newtonian fluids with New-tonian viscosities of η = 1 , . . γ = 1000 s − . The fluorescent nanoparticles used were55 nm-radius (Invitrogen F8803, Thermofisher) and 100nm-radius (Invitrogen F8888 Thermofisher) latex micro-spheres used without further modification besides dilu-tion by a factor of 10 using utra-pure water.TIRFM measurements were realised by illuminatingthe near-surface shear flow with a laser source (CoherentSapphire, wavelength λ = 488 nm, power 150 mW) fo-cused off the central axis of, and on the back focal planeof a 100 × microscope objective with a large numericalaperture (NA = 1 .
46, Leica HCX PL APO). Large NA isrequired to reach incident angles θ larger than the criticalangle, θ c = arcsin( n l /n g ), enabling total reflection at theglass/liquid interface. Here, n g = 1 .
518 is the refractiveindex of the glass coverslip, n f is the refractive index ofthe liquid and θ the angle of incidence of the laser; the re-fractive indices of the three liquids were measured usinga refractometer (Atago PAL-RI).The complete setup and alignment procedure are de-scribed in [50]. In the TIRFM configuration, the evanes-cent wave has an intensity decaying exponentially as I ( z ) = I exp ( − z/ Π) with the penetration depth Π =( λ/ π ) (cid:0) n sin θ − n (cid:1) − / . Typical values of the pen-etration depth were thus Π ≈
100 nm, allowing for anobservation of particles roughly within the first microm-eter from the glass/liquid interface whereas the channelheight is h = 18 µ m. The images of 528 ×
512 pixels (px),with 22.9 px/ µ m, are recorded in 16-bit format (AndorNeo sCMOS) with a frame rate of 400 Hz for a durationof 5 s. For each set of parameters (particle radius, viscos-ity and pressure drop), five videos of 2000 frames wererecorded. Fig. 1(b) shows a superposition of 18 framesshowing a single 55 nm-radius particle’s near-surface tra-jectory in a water flow with an imposed pressure drop of30 mbar across the microfluidic device. After a centroiddetection, the intensity profile was fitted by a radially-symmetric Gaussian model for each frame. Thus the x and y coordinates give the particle position in the planeparallel to the glass/water interface whereas the particleheight, z , is encoded in the intensity thanks to the ex-ponential decay of the evanescent wave. The apparentaltitude is z = Π log ( I /I ) [35, 43]. A discussion linkedto the limitations of using this apparent altitude appearsin the SI.The authors gratefully acknowledge David Lacosteand Andreas Engel for enlightening discussions, andD.L. for technical advice concerning Langevin simula-tions. Patrick Tabeling and Fabrice Monti are likewisethanked for helpful advice related to TIRFM. The au-thors also benefitted from the financial support of CNRS,ESPCI Paris, the Agence Nationale de la Recherche(ANR) under the ENCORE (ANR-15-CE06-005) andCoPinS (ANR-19-CE06-0021) grants, and of the Insti-tut Pierre-Gilles de Gennes (Equipex ANR-10-EQPX-34and Labex ANR-10-LABX- 31), PSL Research Uniersity(Idex ANR-10-IDEX-0001-02). [1] G. I. Taylor, Proceedings of the Royal Society of London.Series A. Mathematical and Physical Sciences , 186(1953).[2] R. Aris, Proceedings of the Royal Society of London.Series A. Mathematical and Physical Sciences , 67(1956).[3] H. Brenner and D. Edwards, Macrotransport Processes,edited by Butterworth (Heinemann, 1993).[4] H. A. Stone and H. Brenner, Industrial & EngineeringChemistry Research , 851 (1999).[5] R. R. Biswas and P. N. Sen, Physical Review Letters ,164501 (2007). [6] I. Griffiths and H. A. Stone, EPL (Europhysics Letters) , 58005 (2012).[7] M. S. Bello, R. Rezzonico, and P. G. Righetti, Science , 773 (1994).[8] C. L. Hansen, M. O. Sommer, and S. R. Quake, Pro-ceedings of the National Academy of Sciences , 14431(2004).[9] T. Van de Ven, Journal of Colloid and Interface Science , 352 (1977).[10] G. Batchelor, Journal of Fluid Mechanics , 369 (1979).[11] R. Foister and T. Van De Ven, Journal of Fluid Mechan-ics , 105 (1980).[12] C. Van den Broeck, J. Sancho, and M. San Miguel, Phys-ica A: Statistical Mechanics and its Applications , 448(1982).[13] K. Miyazaki and D. Bedeaux, Physica A: Statistical Me-chanics and its Applications , 53 (1995).[14] H. Orihara and Y. Takikawa, Physical Review E ,061120 (2011).[15] E. O. Fridjonsson, J. D. Seymour, and S. L. Codd, Phys-ical Review E , 010301 (2014).[16] Y. Takikawa, T. Nunokawa, Y. Sasaki, M. Iwata, andH. Orihara, Physical Review E , 022102 (2019).[17] F. Gentile, M. Ferrari, and P. Decuzzi, Annals of Biomed-ical Engineering , 254 (2008).[18] J. Tan, A. Thomas, and Y. Liu, Soft Matter , 1934(2012).[19] T. K. Nielsen, U. B¨osenberg, R. Gosalawit, M. Dornheim,Y. Cerenius, F. Besenbacher, and T. R. Jensen, ACSNano , 3903 (2010).[20] A. B. Grommet, M. Feller, and R. Klajn, Nature Nan-otechnology , 256 (2020).[21] J. J. Fredberg, Journal of Applied Physiology , 232(1980).[22] J. Grotberg, Annual Review of Fluid Mechanics , 529(1994).[23] L. Salerno, G. Cardillo, and C. Camporeale, Physical Re-view Fluids (2020).[24] D. Stein, F. H. van der Heyden, W. J. Koopmans, andC. Dekker, Proceedings of the National Academy of Sci-ences , 15853 (2006).[25] R. Bearon and A. Hazel, Journal of Fluid Mechanics (2015).[26] S. Marbach, K. Alim, N. Andrew, A. Pringle, and M. P.Brenner, Physical Review Letters , 178103 (2016).[27] S. Marbach and K. Alim, Physical Review Fluids ,114202 (2019).[28] A. Dehkharghani, N. Waisbord, J. Dunkel, and J. S.Guasto, Proceedings of the National Academy of Sciences , 11119 (2019).[29] D. Axelrod, The Journal of cell biology , 141 (1981).[30] K. N. Fish, Current Protocols in Cytometry , 12(2009).[31] D. C. Prieve, Advances in Colloid and Interface Science , 93 (1999).[32] R. Pit, H. Hervet, and L. L´eger, Physical Review Letters , 980 (2000).[33] S. Jin, P. Huang, J. Park, J. Yoo, and K. Breuer, Exper-iments in Fluids , 825 (2004).[34] M. Yoda and Y. Kazoe, Physics of Fluids , 111301(2011).[35] Z. Li, L. D’eramo, C. Lee, F. Monti, M. Yonger, P. Tabel-ing, B. Chollet, B. Bresson, and Y. Tran, Journal of FluidMechanics , 147 (2015). [36] M. Yoda, Annual Review of Fluid Mechanics , 369(2020).[37] P. Huang, J. S. Guasto, and K. S. Breuer, Journal ofFluid Mechanics , 447 (2006).[38] D. Lasne, A. Maali, Y. Amarouchene, L. Cognet, B. Lou-nis, and H. Kellay, Physical Review Letters , 214502(2008).[39] C. Bouzigues, P. Tabeling, and L. Bocquet, Physical Re-view Letters , 114503 (2008).[40] P. Huang and K. S. Breuer, Physical Review E ,046307 (2007).[41] B. Derjaguin, Transactions of the Faraday Society ,203 (1940).[42] E. J. W. Verwey, The Journal of Physical Chemistry ,631 (1947).[43] X. Zheng, F. Shi, and Z. Silber-Li, Microfluidics and Nanofluidics , 127 (2018).[44] L. Korson, W. Drost-Hansen, and F. J. Millero, TheJournal of Physical Chemistry , 34 (1969).[45] A. Einstein, Annalen der Physik , 549 (1905).[46] H. Brenner, Chemical engineering science , 242 (1961).[47] L. P. Faucheux and A. J. Libchaber, Physical Review E , 5158 (1994).[48] A. Saugey, L. Joly, C. Ybert, J.-L. Barrat, and L. Boc-quet, Journal of Physics: Condensed Matter , S4075(2005).[49] D. L. Ermak and J. A. McCammon, The Journal ofChemical Physics , 1352 (1978).[50] M. T. Hoffman, J. Sheung, and P. R. Selvin, in SingleMolecule Enzymology (Springer, 2011), pp. 33–56. upplementary Information for:
Time dependence of advection-diffusion coupling fornanoparticle ensembles
Alexandre Vilquin, Vincent Bertin, Pierre Soulard, Gabriel Guyard,Elie Rapha¨el, Fr´ed´eric Restagno, Thomas Salez and Joshua D. McGraw
Supplementary document information • Video 1: Image sequences of raw experimental data for 55 nm-radius fluorescentparticles advected in a pressure driven flow of pure water along a microchannel withgeometry described in the text, and with an incident 488 nm-laser angle of 64 ◦ .The videos were taken with pressure drops imposed as noted, and each frame has awidth of 23 µ m. • Video 2: Identical to Video 1 but with particle trajectories superimposed. • Video 3: Animations of particle trajectories obtained from Langevin simulations im-posing ‘dot’, ‘half-line’ and ‘line’ initial conditions (indicated by the black regions,see main text for definitions), with diffusion along the vertical and shear flow advec-tion along the horizontal. The animation is displayed during a dimensionless timerange 0 ≤ τ /τ z ≤ . • Video 4: [Left] Identical to Video 3, but in the reference frame of the center of massfor the fluid and for a dimensionless time range 0 ≤ τ /τ z ≤
10. The horizontal black-outlined bar in the middle panel indicates the entire observation zone at τ /τ z = 10,and the observed region at a given instant is highlighted in red. [Right] For thethree different conditions of the right panels, progression of the reduced dispersioncoefficient. • TaylorDispersion.html : Python/Jupyter notebook describing the overdamped Langevinsimulations of Section 7, in particular reproducing Figs. S3 and S4.1 a r X i v : . [ c ond - m a t . s o f t ] J u l Notations used in the main text
Coordinates x streamwisecoordinate, directionof flow y transversecoordinate,flow-invariant z wall-normalcoordinate Times t particle observationtime τ lag time τ z channel explorationtime τ C early-time ‘line-dot’crossover time Diffusioncoefficientsanddispersions D bulk isotropicdiffusion coefficient D y diffusion coefficientin flow-invariantdirection D x dispersion coefficientin the flow direction D h n i dispersion coefficientin the flow directionaveraged over afraction, n of H D τ max D h / i at lateexperimental times F h n i reduceddimensionlessanalogue of D h n i Statisticalquantities P probabilitydistributionfunctions h z i first moment of P ( z ) h z i second moment of P ( z ) σ [] standard deviationof quantity [] Experimentalparameters a particle radius η fluid viscosity ρ fluid density h channel height w channel width ‘ channel length H observation domainsize n fraction of observedflow domain∆ P pressure drop acrosschannel length v x flow speed along the x -direction˙ γ near-wall shear rate U average flow speedin Poiseuille flowRe Reynolds numberPe P´eclet number α Taylor’s geometricdispersion factor λ illumination laserwavelength n f index of refraction:fluid n g index of refraction:glass substrate θ incident laser angle θ c critical incidentangleΠ evanescent wavepenetration length I particle fluorescenceintensity I particle fluorescenceintensity at thesolid/liquid interface2 Intensity distribution and mean velocity profiles
This section provides additional information about how the observed signal intensity dis-tributions (SIDs after Zheng and coworkers [1], denoted P SID ) and the correspondingvelocity profiles can be quantitatively described simultaneously. As also described by Liand coworkers [2], fluorescent nanoparticles detected have a limited range of intensitiesaffected by several factors, the most important ones being, and as discussed in turn: i )electrostatic interactions which determine the probability that a particle of a given radiusis found at a certain distance from the wall according to a Boltzmann distribution; ii )particle size distribution; and iii ) the optical setup which, given the position and size ofthe particle, finally determines its intensity. We now discuss each of these elements indetail.– i – The glass surface exerts an electrostatic repulsion on the particles according to the factthat both surfaces are negatively charged; the details of such a repulsion are understoodwithin the Derjaguin-Landau-Verwey-Overbeek (DLVO) framework [3, 4]. This electro-static interaction potential, φ el , describing the electric double-layer repulsion between aparticle with radius R and a flat wall [5] is given by: φ el ( z c ) = 16 (cid:15)R (cid:18) k B Te (cid:19) tanh (cid:18) eψ p k B T (cid:19) tanh (cid:18) eψ w k B T (cid:19) exp (cid:18) − z c − Rl D (cid:19) . (S1)Here, z c , (cid:15) , e , ψ p , ψ w and l D are respectively the position of the center of the particle, liquidpermittivity, elementary charge, particle and wall electrostatic potentials and the Debyelength. This interaction determines the particle concentration C at thermal equilibriumthrough the Boltzmann distribution C ( z c ) ∝ exp (cid:18) − φ el ( z c ) k B T (cid:19) . (S2)As already observed in TIRFM experiments, the van der Waals interaction can be ne-glected for pure water [1, 2]. Consequently, the typical distance between the bottomsurface (located at z = 0) and the particles is mainly determined by the Debye length.– ii – All the particles do not have the same radius R . The radius distribution is usuallydescribed by a Gaussian probability function P R ( R ) = 1 q πσ R exp − ( R − a ) σ R ! , (S3)where a is the mean radius and σ R its standard deviation.– iii – The fluorescence intensity, I , of an individual particle is determined by the opticalparameters of the TIRFM setup and the particle’s size, with I ∝ R . The evanescentwave has a penetration depth Π characterizing the exponential decrease of excitation.The observed fluorescence intensity is also sensitive to the finite depth of field, d f , ofthe microscope objective. In our experiments, the depth of field has a value of 415 nm,meaning that if particles are not located on the focal plane at z f (typically 400-500 nm fromthe glass-liquid interface), they will be detected with a relatively low intensity. Puttingthese elements together, the observed fluorescence intensity for an individual particle ispredicted [1] as I ( R, z c ) I = (cid:18) Ra (cid:19) exp (cid:18) − z c − R Π (cid:19) " (cid:18) z c − R − z f d f (cid:19) − , (S4)where I is the intensity for a particle with radius R = a located at the bottom surface z c = a and with the focal plane at the wall ( z f = 0).Using a home-made Matlab interface, we combine Eqs. S1-S4 to generate theoreticalSIDs numerically. Practically, we determine the fraction of particles having an altitude z c (a) Comparison between experimental and theoretical signal intensity distri-butions (SIDs). (b) Comparison between experimental and theoretical streamwise meanvelocity profiles. The experimental data is for 55 nm-radius particles for a pressure dropof 30 mbar across the microchannel. and a radius R given by the weight W ( z c , R ) = C ( z c ) P R ( R ), and compute the associatedintensity given by Eq. S4. This procedure gives a list of weighted intensities formingthe blue line shown in Fig. S1(a) using a DLVO prefactor 16 a(cid:15) ( k B T /e ) tanh( e Ψ p / k B T )tanh( e Ψ w / k B T ) = 1 . × − J, l D = 60 nm, σ R = 5 . a = 55 nm and the opticalparameters as described above, along with the experimental histogram (red).In addition, we also quantitatively describe the mean streamwise velocity profile, which,as noted before [2], is not perfectly linear when using the apparent altitude z = z c − a =Π ln ( I /I ) defined in the main article. We assume that a particle located at an altitude z c has a mean streamwise velocity v x given by v x ( z ) = f B ˙ γz , where ˙ γ is the shear rate and f B the “Brenner factor” [6]. This factor provides the hydrodynamic correction induced by thefinite size of the spherical particle, when the latter is advected by a linear shear flow near awall. For large z c /R , the Brenner factor can be expressed as f B ’ − (5 /
16) ( z c /R ) − . For55 nm-radius particles typically located at distances larger than 200 nm due to electrostaticrepulsion, the deviation from the linear velocity profile is less than 1%.Using the proposed particle velocity profile in conjunction with the intensity-altitude-probability relations (Eqs. S1-S4), we follow Zheng et al. [1] and predict the particle’smean streamwise velocity as a function of log( I /I ). Such a prediction is made with ˙ γ adjusted simultaneously to the physical and optical parameters of Eqs. S1-S4. The result isshown together with the experimental results in Fig. S1(b), showing good agreement andcapturing the main nonlinear features of the experimental data. The shear-rate valuesobtained with this SID method are approximately 15% smaller than the ones directlyobtained using a linear regression of the velocity profiles of Fig. 1(c) using the apparentaltitude. This discrepancy is mainly due to the particle polydispersity and to the finitedepth of field of the microscope objective, and since it is only a constant factor (verified)across all experiments it does not change the main conclusions of the article. In Fig. 1(c), we show the streamwise velocity profiles for 55 nm-radius particles in awater flow obtained by total internal reflection fluorescence microscopy (TIRFM). In thecorresponding inset, we show the associated shear rate ˙ γ (obtained from a linear regressionon a given velocity profile) as a function of the pressure drop ∆ P across the channel.Similar measurements were done for the 100 nm-radius particles in water and water-glycerol mixtures presented in the main article, see Figs. 3 and 4. Corresponding to theseshear rate measurements, we can compute the stress, Σ = h ∆ P/ ‘ , from the pressure4igure S2: (a) Comparison for stresses Σ versus shear rate ˙ γ between TIRFM (circles)and rheology measurements (lines) for 100 nm-radius particles, see inset of Fig. 1(c) for a = 55 nm particles in water. For TIRFM, the stress is calculated as Σ = h ∆ P/ ‘ where ∆ P is the pressure gap in the channel, h and ‘ are the channel height and length. (b) Bulkdiffusion coefficient D , measured from the plateau values of local transverse mean squaredisplacements, versus the theoretical values calculated using the viscosity measured with arheometer, for all particle sizes and water-glycerol mixtures. The black dashed line showsthe linear relation with unit prefactor. drop across the rectangular channel using a geometric prefactor (height h = 18 µ m, width w = 180 µ m, length ‘ = 8 . γ in Fig. S2(a) and compared with bulk rheology measurements carried out in a Couette cell(see Methods). First, the resulting linear power laws show that all the solutions remainNewtonian for shear rates up to 1000 s − . Second, the viscosity defined as η = Σ / ˙ γ isconsequently constant for a given solution, and it increases with the glycerol proportion.The results show a good agreement with the rheology measurements, validating both theshear rate and viscosity values obtained by TIRFM.The viscosity values obtained further allow us to verify that the bulk diffusion coeffi-cients measured with TIRFM are consistent with the Stokes-Einstein relation. As shownin Fig. 3(a) in the main article, the bulk diffusion coefficient D is obtained from theplateau value of the local transverse diffusion coefficient D y , calculated from the trans-verse mean-square displacement σ y , through: D y = σ y / τ , where τ is the lag time.In Fig. S2(b) is shown a comparison between the experimental results and the predictiongiven by the Stokes-Einstein relation [7]: D = k B T / πηa , after having used the inde-pendent, rheologically measured viscosity. The good agreement validates the statisticalmethod to obtain the bulk diffusion coefficient. Here, we justify the expression (see Eq. 1 of the main article) of the dispersion coefficientfor a linear shear flow in the long-time limit: D x = D (cid:18) γ H D (cid:19) , τ (cid:29) τ z . (S5)The classical Taylor dispersion coefficient is typically calculated for a channel with eithera circular or a rectangular section and for a Poiseuille flow [8]. To interpret the resultsdescribed in Figs. 3 and 4 in the main article, we revisit its derivation for a linear shear flow.First, we consider a population of identical spherical colloids, in a rectangular channel ofheight H , subjected to a linear shear flow v x ( z ) = ˙ γz along the x -axis, where ˙ γ is a constant5hear rate and where 0 z H . Thus, the mean velocity along x is H R H v x ( z ) d z =˙ γH/
2. We assume invariance in the y -direction which is valid in the experiments. We alsoneglect hydrodynamic interactions with the walls, which would introduce a z -dependenceof the diffusion coefficients in both the streamwise and wall-normal directions. Therefore,all the colloids are assumed to have the same, bulk diffusion coefficient D . The colloidalconcentration field c ( x, z, t ) evolves with time t , from both advection by the imposed flowand diffusion, as described by the advection-diffusion equation: ∂ t c + v x ∂ x c = D (cid:0) ∂ x c + ∂ z c (cid:1) . (S6)Introducing the streamwise length scale L , a concentration scale c , the dimensionless vari-ables Z = z/H , X = x/L , T = tD /H , C ( X, Z, T ) = c ( x, z, t ) /c , V ( Z ) = v x ( z ) / ˙ γH ,the P´eclet number Pe = ˙ γH /D and the aspect ratio ε = H/L , Eq. (S6) becomes: ∂ T C + ε Pe V ( Z ) ∂ X C = ε ∂ X C + ∂ Z C. (S7)We decompose the concentration profile through: C ( X, Z, T ) = ¯ C ( X, T ) + C ( X, Z, T ) , (S8)with ¯ C ( X, T ) = R C ( X, Z, T ) d Z the thickness-averaged concentration and C the devi-ation from the latter. Inserting this decomposition into Eq. (S7), one obtains: ∂ T ¯ C + ∂ T C + ε Pe V ( Z ) ∂ X ¯ C + ε Pe V ( Z ) ∂ X C = ε ∂ X ¯ C + ε ∂ X C + ∂ Z C . (S9)Averaging further over Z leads to: ∂ T ¯ C + ε Pe 12 ∂ X ¯ C + ε Pe V ∂ X C = ε ∂ X ¯ C + ε ∂ X C , (S10)where we assumed no normal colloidal flux at the channel boundary. Subtracting the twolast equations with one another gives: ∂ T C + ε Pe (cid:18) V − (cid:19) ∂ X ¯ C + ε Pe (cid:0) V ∂ X C − V ∂ X C (cid:1) = ε (cid:16) ∂ X C − ∂ X C (cid:17) + ∂ Z C . (S11)Invoking now the long-time condition proposed by Taylor [8], T (cid:29)
1, the concentrationfield becomes nearly homogeneous along Z leading to C (cid:28) ¯ C . In addition, we have ε (cid:28) ∂ Z C ’ ε Pe (cid:18) V − (cid:19) ∂ X ¯ C. (S12)Since the thickness-averaged concentration field ¯ C does not depend on Z , a first integrationgives: ∂ Z C = ε Pe ∂ X ¯ C (cid:18) Z − Z (cid:19) + c , (S13)where c is a constant. Due to the impermeability of the channel walls, the no-fluxboundary conditions are ∂ Z C | Z =0 = 0 and ∂ Z C | Z =1 = 0 giving c = 0. Thus, the secondintegration gives: C = ε Pe ∂ X ¯ C (cid:18) Z − Z c (cid:19) . (S14)By definition, the thickness average of the deviation field C is zero, leading to: C = ε Pe ∂ X ¯ C (cid:20) Z − Z (cid:21) . (S15)6nvoking the found relation between C and ¯ C , Eq. (S10) becomes: ∂ T ¯ C + 12 ε Pe ∂ X ¯ C = ε (cid:0) B Pe (cid:1) ∂ X ¯ C, (S16)with B a dimensionless factor given by: B = − Z d Z Z (cid:18) Z − Z (cid:19) = 1120 . (S17)Putting back the dimensions and adding the definition ¯ c = c ¯ C , we obtain the finalequation for the long-time limit: ∂ t ¯ c + ˙ γH ∂ x ¯ c = D x ∂ x ¯ c, (S18)with the dispersion coefficient D x = D (cid:0) B Pe (cid:1) where B = 1 / γH/ D x = D (cid:18) γ H D (cid:19) , τ (cid:29) τ z . (S19) Here, we justify that the relaxation of an initial concentration profile in a rectangularchannel takes place over the time scale τ z /π where the Taylor time is τ z = H /D .This quantity corresponds to the time beyond which the dispersion coefficient, Eq. (S19),calculated above becomes valid. According to Taylor [8], this time scale correspondsadditionally to the duration needed to have a homogenous concentration field along z .This phenomenon is only due to the diffusion along z , leading us to solve: ∂ t c = D ∂ z c, (S20)where c ( z, t ) is the concentration field, the other spatial dependencies being irrelevant. Weassume an initial concentration field c ( z, t = 0) = c ( z ) and solve the diffusion equationon a domain z ∈ [0 , H ] along with impermeability boundary conditions: ∂ z c ( z, t ) = 0 at z = 0 and z = H . This problem can be solved exactly using the spectral decomposition: c ( z, t ) = ¯ c + ∞ X k =1 a k ( t ) cos (cid:18) kπzH (cid:19) , (S21)where ¯ c denotes the thickness-averaged concentration, and the coefficients a k follow thelinear ordinary differential equations: ∂ t a k = − D ( kπ/H ) a k . The general solution of thisdiffusion problem is c ( z, t ) = ¯ c + ∞ X k =1 a k, cos (cid:18) kπzH (cid:19) exp (cid:18) − k π D tH (cid:19) , (S22)where a k, = a k (0) = (2 /H ) R H c ( z ) cos( kπz/H ) d z . As a result, the slowest decayingmode k = 1 has a typical decay time H /π D = τ z /π as desired.7 Taylor dispersion in a linear shear flow at short times fordifferent initial particle distributions: short times
In this section, we first provide the derivation of the short-term dispersion coefficientfor a general distribution of initial particle altitudes in free space, before addressing theparticular case of uniform distributions. A central assumption of the model presentedin the main article is that the short-time Taylor dispersion in a channel, behaves in thesame way as for an infinitely-extended medium. It is supported by the fact that the wall-normal diffusion of a particle in the channel is not affected by the presence of the wallsat short times. Therefore, this assumption is valid only for time scales much shorter thanthe typical time τ z = H /D for diffusion over the channel height, as introduced in theprevious section. In this subsection, we justify how the short-time dispersion coefficient for a unique particlein a linear shear flow [9, 10, 11, 12, 13] D x = D (cid:0) γ τ / (cid:1) , is modified by consideringa group of particles leaving from different altitudes z . The single-particle expression,analogous to the ‘dot’ condition in the main article, is neither valid in our experimentsnor in the general case. Thus, we consider particles that are advected in a shear flow afterhaving started at x = 0, from different initial altitudes z described by the probabilitydensity function (PDF) P ( z ). The particles are assumed to diffuse in an infinite spacealong z . We stress that the hydrodynamic and electrostatic interactions with the wall,which are present in the experiments, are neglected here. We will discuss their effects onthe Taylor dispersion in the next section, using Langevin simulations.The calculation of the dispersion coefficient D x = σ x / τ requires the variance σ x = (cid:10) ∆ x (cid:11) −h ∆ x i of the streamwise displacement ∆ x ( τ ) = x ( t + τ ) − x ( t ). We assume here alinear shear flow, with a velocity v x ( z ) = ˙ γz along x . We do not consider the streamwiseBrownian motion since it is not correlated to the advection and can be superimposedafterwards by linearity. Besides, the experimental time scale is much larger than thetypical cross-over time between ballistic and diffusive motion (approximately 1 ns) so thatwe consider the overdamped dynamics. Therefore the governing equation in the streamwisedirection is ∂ t x = ˙ γz . Upon integration, one gets:∆ x ( τ ) = ˙ γ τ Z d t z ( t ) . (S23)The mean value of the displacement is thus: h ∆ x ( τ ) i = ˙ γ τ Z d t h z ( t ) i . (S24)In this model, the vertical motion is purely Brownian and is described by the overdampedLangevin equation: ∂ t z = √ D ξ ( t ), where ξ ( t ) is a Gaussian white noise. The corre-sponding one-dimensional Brownian propagator is:P z ( z, t | z , t ) = 1 p πD ( t − t ) exp " − ( z − z ) D ( t − t ) . (S25)P z ( z, t | z , t ) represents the density of probability per unit length for the particle to belocated at altitude z at time t , under the condition that the particle was located at z attime t . From P z and Eq. (S24), one obtains h ∆ x i = ˙ γ h z i τ .8oving to the calculation of the second moment, we have the general form: (cid:10) ∆ x ( τ ) (cid:11) = ˙ γ τ Z d t τ Z d t h z ( t ) z ( t ) i , (S26)which can be calculated from the propagator P z . We note that the averaged quantity h z ( t ) z ( t ) i is the product of the particle altitude at time t and the particle altitude attime t for a unique trajectory, and consequently: h z ( t ) z ( t ) i 6 = Z d z Z d z z P z ( z , t | z , t ) z P z ( z , t | z , t ) . (S27)Assuming t < t , the correct expression is provided by the Markovian properties ofBrownian motion and reads: h z ( t ) z ( t ) i = Z d z Z d z z P z ( z , t | z , t ) z P z ( z , t | z , t ) . (S28)Knowing that R d z z P z ( z , t | z , t ) = z and choosing t = 0, we have: h z ( t ) z ( t ) i = Z d z z P z ( z , t | z , t ) = z + 2 D t . (S29)Without the assumption t < t , the latter expression is generalized as: h z ( t ) z ( t ) i = z + 2 D min ( t , t ) , (S30)and we note here that t = t = t leads to the classical result (cid:10) z ( t ) (cid:11) − h z ( t ) i = 2 D t .In the case of an initial assembly of spatially-distributed identical particles, accordingto the PDF P ( z ), the latter equation is replaced by its average over z : h z ( t ) z ( t ) i = (cid:10) z (cid:11) + 2 D min ( t , t ) . (S31)By inserting this expression in Eq. (S26), one gets: (cid:10) ∆ x ( τ ) (cid:11) = ˙ γ τ Z d t τ Z d t (cid:2)(cid:10) z (cid:11) + 2 D min ( t , t ) (cid:3) . (S32)Invoking the decomposition R τ d t = R t d t + R τt d t , it follows: (cid:10) ∆ x ( τ ) (cid:11) = ˙ γ (cid:10) z (cid:11) τ Z d t τ Z d t | {z } τ +2 ˙ γ D τ Z d t (cid:18) t Z d t t | {z } t / + τ Z t d t t | {z } ( τ − t ) t (cid:19) , (S33)and thus: (cid:10) ∆ x ( τ ) (cid:11) = ˙ γ (cid:10) z (cid:11) τ + 23 ˙ γ D τ . (S34)From the above expressions of the average and mean-squared displacements, and addingfurther the independent contribution due to streamwise Brownian motion, the variance ofthe streamwise displacement becomes: σ x ( τ ) = 2 D τ + ˙ γ (cid:16)(cid:10) z (cid:11) − h z i (cid:17) τ + 23 ˙ γ D τ . (S35)Therefore, the dispersion coefficient can be finally expressed as: D x = D + ˙ γ (cid:10) z (cid:11) − h z i τ + 13 ˙ γ D τ . (S36)9he quadratic dependance in lag time τ predicted by the term ˙ γ D τ / γ (cid:16)(cid:10) z (cid:11) − h z i (cid:17) τ /
2. This term is linear in lag time and dominates for time scales smallerthan the crossover time τ C = 3 (cid:16)(cid:10) z (cid:11) − h z i (cid:17) / D , which corresponds to a typical timeneeded to diffuse over the standard deviation of the initial distribution P in altitudes z . We focus now on the particular case of uniform distributions of initial particle altitudes. Inpractice, we specify further the central term ˙ γ (cid:16)(cid:10) z (cid:11) − h z i (cid:17) τ / D x = D (cid:0) γ τ / (cid:1) , isto consider that all the particles leave from a unique initial altitude z i . This correspondsto the PDF P ( z ) = δ ( z − z i ), where δ is the Dirac distribution. From the latter, itfollows that h z i = (cid:10) z (cid:11) = z , leading to the vanishing of the linear term in lag time inthe dispersion coefficient.A more general situation arises by considering that the particles are initially uniformlydistributed over a vertical segment of length nH and centred at altitude z i , where n isthe dimensionless fraction of the typical vertical length H . Note that, when comparingwith experimental data (see Fig. 4 in the main article), H denotes the thickness of theobservation zone. This situation is described by the PDF: P ( z ) = (cid:26) | z − z i | > nH/ /nH if | z − z i | nH/ . (S37)Thus, the additional term ˙ γ (cid:16)(cid:10) z (cid:11) − h z i (cid:17) τ /2 due to the initial distribution of altitudes,can be calculated explicitly by expressing the average initial altitude: h z i = H Z d z P ( z ) z = 1 nH z i + nH/ Z z i − nH/ d z z = z i , (S38)and the variance in initial altitude: (cid:10) z (cid:11) = H Z d z P ( z ) z = 1 nH z i + nH/ Z z i − nH/ d z z = z + ( nH ) . (S39)Consequently, Eq. (S36) becomes: D x ( τ ) = D + ˙ γ ( nH ) τ + 13 ˙ γ D τ , (S40)where the weight of the linear term in lag time depends solely on the spatial extent of theinitial distribution, but not on the average initial altitude z i itself. Note that this bulkresult is expected to be modified in presence of confinement and interfacial effects. Finally,by invoking the time scale τ z = H /D as in the main article, we obtain the dimensionlessequation: (cid:18) D x D − (cid:19) ( ˙ γτ z ) − = n ττ z + 13 (cid:18) ττ z (cid:19) . (S41)We recall that this expression is valid for short lag times τ compared to the Taylor time τ z /π . Interestingly, this single expression allows us to consider various initial uniformdistributions, from the classical “dot condition” ( n = 0) to a complete “line condition”( n = 1). As shown in the main article, this equation is in good agreement with ourexperiments in the short-time limit. 10 Simulation of Langevin equations
In this section, we provide details on the simulations of the Langevin equations, corre-sponding to the results shown in the main article (see Fig. 4) for comparison with ourexperimental results and with the analytical models described in the previous sections.
For a single Brownian particle of position coordinates x i , with i the coordinate index,advected in an external flow field characterized by the fluid velocity components v i , thediscrete overdamped Langevin equations read in the Ito convention [17]: x i ( t + δt ) = x i ( t ) + v i δt + D i ( t ) F i ( t ) k B T δt + ∂ x i D i | x i ( t ) δt + p D i ( t ) δt S (0 , , (S42)where the diffusion coefficients D i are non-isotropic and space-dependent in the generalcase, as a consequence of hydrodynamic interactions with the walls. Here, k B denotes theBoltzmann constant, T the absolute temperature, S ( m, σ ) a Gaussian distribution withmean value m and standard deviation σ , and F i the components of the external forceexerted on the particle.More specifically, we consider the bidimensional problem of a Brownian particle ad-vected along x by a linear shear flow near a wall. Gravity is neglected such that the onlyexternal force considered is the electrostatic force F el exerted by the wall in the normaldirection z . The fluid velocity profile is given by v x ( z ) = ˙ γz , with ˙ γ a constant shear rate.Eq. (S42) thus becomes: x ( t + δt ) = x ( t ) + ˙ γz ( t ) δt + p D x ( z ( t )) δt S (0 , , (S43) z ( t + δt ) = z ( t ) + ∂ z D z | z ( t ) δt + D z ( z ( t )) F el ( z ( t )) k B T δt + p D z ( z ( t )) δt S (0 , . (S44)For a particle with a radius R , the bulk diffusion coefficient is given by D = k B T / πηR ,with η the dynamic shear viscosity of the liquid. We introduce the dimensionless variable Z = z/R . Due to the hydrodynamic interactions with the wall, the streamwise and wall-normal diffusion coefficients are modified as D i = D β i [18, 2], with: β x ( Z ) = 1 − Z − + 18 Z − − Z − − Z − + O (cid:0) Z − (cid:1) , (S45) β z ( Z ) = 6( Z − + 2( Z − Z − + 9( Z −
1) + 2 . (S46)Finally, the electrostatic force is given by F el = − ∂ z φ el , where φ el is the repulsive electro-static potential due to the double layer [1] (see section 2). In the simulations, the van derWaals interactions are neglected. We numerically integrate Eqs. (S43) and (S44) with a home-made
Python program, seethe associated Jupyter notebook (
TaylorDispersion.html ). In order to discuss the mainassumption of section 6, we first focus on the simple case of Brownian tracer particles ( i.e. β x ( Z ) = β z ( Z ) = 1) that are advected in a linear shear flow, within in a closed channel ofthickness H , and in the absence of any electrostatic repulsion ( F el = 0). We generate 10 particle trajectories leaving from initial altitudes described by the uniform distributionsdefined in the previous section. As such, we consider that the particles are initially locatedon a line of spatial extent nH . The numerical time step is typically δt = 0 . τ z , and thetotal duration is approximately 10 τ z . Reflective boundary conditions are used at the twowalls, i.e. at z = 0 and z = H . The simulations are performed for the dot ( n = 0), half-line11igure S3: Reduced dispersion coefficient as a function of dimensionless lag time, fromLangevin simulations in free space (circles) and in a confined channel (diamonds), fordifferent initial conditions: dot (blue), half-line (red) and line (green). The coloured plainlines and black dashed line respectively show the theoretical predictions for the short-time(Eq. (S41) ) and long-time (Eq. (S19) ) limits. ( n = 1 /
2) and line ( n = 1) conditions. To highlight the relative effect of confinement onthe dispersion, we also performed reference simulations in free space ( i.e. without walls).In all cases, the dispersion coefficient D x is calculated from the mean-squared displacement(see section 6) and shown in Fig. S3.For all conditions, the free-space reduced dispersion coefficient continuously increaseswith time and is in agreement at all times with the predictions of Eq. (S41). In particular,we clearly observe the expected crossover from the linear to the quadratic behaviours inlag time for particles that are initially broadly distributed in altitudes ( n = 1).Furthermore, the Langevin simulation results of the Taylor dispersion in a finite-sizedchannel are also plotted in Fig. S3. The reduced dispersion coefficients are found to firstincrease at times shorter than τ z , following Eq. (S41) as in the free-space case. In contrastto the latter, in the long-time limit, the reduced dispersion coefficient saturates and reachesthe constant value predicted by Eq. (S19) no matter the initial particle distribution. The previous simulations have been performed without hindered diffusion and electrostaticinteractions, which are physical effects that may impact the experiments and that areinherent to nanofluidic settings. We thus performed additional Langevin simulations,including the interactions between the finite-size colloids and the glass wall located at z =0. Simple reflective boundary conditions are maintained at the other boundary, locatedat z = H . The electrostatic parameters were chosen as those that give agreement betweenthe experimental and theoretical SID. Specifically, the particle-wall interaction, φ el , isexponentially decaying with magnitude 1 . × − J, Debye length 60 nm and thermalenergy 4 . × − J for input to Eq. S1. Furthermore, at the initial time, the particlesare placed in the segments (dot, half-line, line) of the previous section also following theBoltzmann distribution C ( z c ) ∝ exp( − φ el ( z c ) /kT ). As described by Eqs. (S45) and (S46),we also incorporate non-trivial, hydrodynamic interactions with the wall.In Fig. S4, we display in red the reduced dispersion coefficients obtained from theLangenvin simulations incorporating electrostatic and hydrodynamic interactions with thewall. These are shown as a function of dimensionless time, using τ z = H /D , for all initialconditions. The rescaled dispersion coefficients are found to be systematically smaller thanthe ones for tracer molecules in finite-sized channels.12igure S4: Reduced dispersion coefficient from (red) Langevin simulations in a channel,including electrostatic repulsion and hindered diffusion induced by the wall located at z = 0 .These are plotted for three different initial conditions (a,b,c) as a function of dimensionlesstime. For comparison, we plot the same data (orange) with time on both axes rescaledby the modified time scale τ z = (0 . H ) /D , and the simulation data of Fig. S3 for(blue) tracer particles in a channel with no interactions with the wall. The lines indicatethe asymptotic limits for (horizontal dashed) long times and short times (sloping solid);respectively, a constant value of 1/120 and Eq. 3 in the main paper. To justify the difference between tracers and finite-sized, interacting particles, we par-ticularly note that electrostatic forces repel the colloids from the wall, and therefore re-duce the area accessible to them. As a secondary effect, the hydrodynamic interactionsare thus also reduced since those are maximal at the wall; we will thus assume in thefollowing that D is not modified even while the effect may be slightly operative. As aresult of the effectively reduced channel size, the associated timescale should be modified.Therefore, we replot in Fig. S4 the reduced dispersion coefficients versus the dimensionlesstime by modifying τ z , using rather 0 . H , giving a smaller, empirical diffusion time scale τ z = (0 . H ) /D . We find that the simulation results in the presence of interactionswith the wall in an effectively smaller channel of size 0 . H agree well with the simulationresults for tracer particles in the original channel of size H , whatever the initial altitudedistributions. This result indicates that the combined effect of electrostatic interactionsand hindered diffusion mainly lead – in our experimental range – to a reduced effectivetime scale τ z without significantly altering the time dependence itself. References [1] Xu Zheng, Fei Shi, and Zhanhua Silber-Li. Study on the statistical intensity dis-tribution (sid) of fluorescent nanoparticles in tirfm measurement.
Microfluidics andNanofluidics , 22(11):127, 2018.[2] Zhenzhen Li, Lo¨ıc Deramo, Choongyeop Lee, Fabrice Monti, Marc Yonger, PatrickTabeling, Benjamin Chollet, Bruno Bresson, and Yvette Tran. Near-wall nanove-locimetry based on total internal reflection fluorescence with continuous tracking.
Journal of Fluid Mechanics , 766:147–171, 2015.[3] B Derjaguin. On the repulsive forces between charged colloid particles and on thetheory of slow coagulation and stability of lyophobe sols.
Transactions of the FaradaySociety , 35:203–215, 1940.[4] Evert Johannes Willem Verwey. Theory of the stability of lyophobic colloids.
TheJournal of Physical Chemistry , 51(3):631–636, 1947.[5] Dennis C Prieve. Measurement of colloidal forces with tirm.
Advances in Colloid andInterface Science , 82(1-3):93–125, 1999.136] AJ Goldman, RG Cox, and H Brenner. Slow viscous motion of a sphere parallel to aplane wallii couette flow.
Chemical Engineering Science , 22(4):653–660, 1967.[7] Albert Einstein. ¨Uber die von der molekularkinetischen theorie der w¨arme gefordertebewegung von in ruhenden fl¨ussigkeiten suspendierten teilchen.
Annalen der physik ,322(8):549–560, 1905.[8] Geoffrey Ingram Taylor. Dispersion of soluble matter in solvent flowing slowly througha tube.
Proceedings of the Royal Society of London. Series A. Mathematical andPhysical Sciences , 219(1137):186–203, 1953.[9] TGM Van de Ven. Diffusion of brownian particles in shear flow.
Journal of Colloidand Interface Science , 62(2):352–355, 1977.[10] GK Batchelor. Mass transfer from a particle suspended in fluid with a steady linearambient velocity distribution.
Journal of Fluid Mechanics , 95(2):369–400, 1979.[11] RT Foister and TGM Van De Ven. Diffusion of brownian particles in shear flows.
Journal of Fluid Mechanics , 96(1):105–132, 1980.[12] C Van den Broeck, JM Sancho, and M San Miguel. Harmonically bound brown-ian motion in flowing fluids.
Physica A: Statistical Mechanics and its Applications ,116(3):448–461, 1982.[13] Kunimasa Miyazaki and Dick Bedeaux. Brownian motion in a fluid in simple shearflow.
Physica A: Statistical Mechanics and its Applications , 217(1-2):53–74, 1995.[14] Hiroshi Orihara and Yoshinori Takikawa. Brownian motion in shear flow: Directobservation of anomalous diffusion.
Physical Review E , 84(6):061120, 2011.[15] Einar Orn Fridjonsson, Joseph D Seymour, and Sarah L Codd. Anomalous preasymp-totic colloid transport by hydrodynamic dispersion in microfluidic capillary flow.
Physical Review E , 90(1):010301, 2014.[16] Yoshinori Takikawa, Takahiro Nunokawa, Yuji Sasaki, Makoto Iwata, and HiroshiOrihara. Three-dimensional observation of brownian particles under steady shearflow by stereo microscopy.
Physical Review E , 100(2):022102, 2019.[17] Donald L Ermak and J Andrew McCammon. Brownian dynamics with hydrodynamicinteractions.
The Journal of chemical physics , 69(4):1352–1360, 1978.[18] Luc P Faucheux and Albert J Libchaber. Confined brownian motion.