Particle sizing for flowing colloidal suspensions using flow-differential dynamic microscopy
aa r X i v : . [ c ond - m a t . s o f t ] F e b Particle sizing for flowing colloidal suspensions usingflow-differential dynamic microscopy † James A. Richards, ∗ Vincent A. Martinez, ∗ and Jochen Arlt ∗ Particle size is a key variable in understanding the behaviour of the particulate products that underpinmuch of our modern lives. Typically obtained from suspensions at rest, measuring the particle sizeunder flowing conditions would enable advances for in-line testing during manufacture and high-throughput testing during development. However, samples are often turbid, multiply scattering lightand preventing the direct use of common sizing techniques. Differential dynamic microscopy (DDM)is a powerful technique for analysing video microscopy of such samples, measuring diffusion andhence particle size without the need to resolve individual particles while free of substantial user input.However, when applying DDM to a flowing sample, diffusive dynamics are rapidly dominated byflow effects, preventing particle sizing. Here, we develop “flow-DDM”, a novel analysis scheme thatcombines optimised imaging conditions, a drift-velocity correction and modelling of the impact offlow. Flow-DDM allows a decoupling of flow from diffusive motion that facilitates successful particlesize measurements at flow speeds an order of magnitude higher than for DDM. We demonstratethe generality of the technique by applying flow-DDM to two separate microscopy methods and flowgeometries.
Solid micron-sized particles, say from
100 nm to several µ m , dis-persed in a liquid are ever present in our lives. These col-loidal suspensions form the basis of consumer formulations ( e.g. sunscreen), construction materials, and even pharmaceuticals orfood. In all these applications the particle size can be of criticalimportance for performance, controlling the strength of concrete and paint film formation, or the rates of drug adsorbtion. Par-ticle size can even influence our sensory perception of materials,as with the taste of chocolate. Measuring the size of particles in formulations is therefore animportant task, both during development ( e.g. high-throughputtesting), but also in real-time during manufacture to ensure aconsistent formulation. To achieve these goals it is necessary tocharacterise a suspension not just in a quiescent (non-flowing)state but also under flowing conditions. For quiescent samples,various approaches to particle sizing exist, for which the refer-ence method is to determine size directly from high-resolutionelectron microscopy images. However, this requires dry parti-cles (it is not an in situ method) and expensive equipment. Moreroutine laboratory techniques for sizing particles in suspensioninclude the well-established methods of static and dynamic light-scattering (SLS and DLS). SLS measures the particle form fac-tor (and hence size) from the average intensity scattered; in con-trast, DLS measures the free diffusion coefficient, D , via tempo-ral fluctuations of the scattered intensity due to Brownian motion. SUPA and School of Physics and Astronomy, University of Edinburgh, King’s Build-ings, Edinburgh EH9 3FD, United Kingdom. E-mail: [email protected],[email protected], [email protected] † Electronic Supplementary Information (ESI) available: containing details on theimpact of the form of the residual velocity distribution, depth dependence, q -dependent fitting, extracted velocity distributions and far-field correlator results. From D , the particle diameter, d , can be extracted via the Stokes-Einstein relation. DLS has been extended to flowing systems forin-line testing, but the measured particle size is impacted by flowspeed. However, for formulation science a more fundamental issuearises for both SLS and DLS, as the techniques are strongly af-fected by multiple scattering, where photons interact with morethan one particle before reaching the detector. Although suppres-sion of multiple scattering is possible using advanced DLS tech-niques, highly dilute and transparent samples are required forstandard commercial DLS setups. For formulations, which mayeven be turbid, this is an excessively restrictive requirement.This limitation arises from the fact that DLS operates on a largescattering volume. One can also extract size from dynamics ina smaller volume by tracking individual particles from video mi-croscopy. However, this approach requires identifying individ-ual particles, a task which becomes impracticable for smaller par-ticles ( d .
500 nm ) or in non-dilute, turbid systems, althoughone which machine learning is being applied to. Using differ-ential dynamic microscopy (DDM) , a digital Fourier analysis ofvideo microscopy, we avoid both user inputs and particle location.DDM has been used to characterise the micro-rheological prop-erties of fluids; to enable high-throughput measurements ofmicro-organism motility; and to measure particle diffusionin complex environments, under external fields, and evenin dense or turbid systems. However, for flowing suspensions the fluid’s velocity can impactmany particle-sizing techniques, causing an apparent increase indiffusion and an underestimation of particle size.
Therefore,for reliable particle sizing microscopic diffusive motion must bedisentangled from the impact of bulk flow. The effect of flow onanother digital Fourier microscopy technique related to DDMhas recently been suggested, but this was limited to exploring ualitative changes in the microscopic dynamics of soft solids. Here, we present “flow-DDM”, a novel DDM-based analysisscheme to quantitatively measure diffusive dynamics in flowingsamples using a combination of drift-velocity correction and anappropriate theoretical model. Respectively, these reduce thecontribution of the flow to the dynamics and allow a careful de-coupling of the diffusive dynamics from the residual effects offlow. Using dilute colloidal suspensions, we systematically vali-date flow-DDM as a function of flow speed for the accurate mea-surement of particle size. We find that flow-DDM outperformscurrent DDM techniques by an order of magnitude in the max-imum possible flow speed. We establish a measurement proto-col, bounds for reliable diffusion measurements and a guide tooptimise the imaging method, which together could be widely ap-plied for particle sizing in a multitude of flowing samples. This isdemonstrated using phase-contrast microscopy of Poiseuille flowand fluorescence confocal microscopy of a rheometric shear flow.
Before describing the impact of flow, we shall briefly cover howDDM can be used for particle sizing in a quiescent sample. DDMcharacterises the spatio-temporal density fluctuations within asample by analysing microscopy movies, I ( ~ r , t ) , of a sample re-gion. Specifically, one computes the differential intensity correla-tion function (DICF), also known as the image structure function: g ( ~ q , τ ) = D | ˜ I ( ~ q , t + τ ) − ˜ I ( ~ q , t ) | E t (1)with ˜ I ( ~ q , t ) the Fourier transform of I ( ~ r , t ) and τ the delay time.Under appropriate imaging conditions and assuming the inten-sity fluctuations are proportional to fluctuations in sample density( ∆ I ∝ ∆ρ ) the DICF can be written as g ( ~ q , τ ) = A ( ~ q )[ − ℜ ( f ( ~ q , τ ))] + B ( ~ q ) , (2)where A ( ~ q ) characterises the signal amplitude (which will dependboth on sample properties, such as the particle’s form factor, andthe imaging system) and B ( ~ q ) accounts for uncorrelated back-ground noise. Here f ( ~ q , τ ) , often known as the intermediate scat-tering function (ISF), is the ~ q -Fourier component of the probabil-ity of the particle displacements, δ ~ r = ~ r j ( t + τ ) − ~ r j ( t ) , f ( ~ q , τ ) = h e i ~ q · δ ~ r i j , t , (3)with brackets denoting averages over all particles j and time t . Inthe absence of net flow, f ( ~ q , τ ) is a real valued function and ifthe underlying dynamics are isotropic it only depends on | ~ q | = q ,leading back to the more familiar, simplified expression: g ( q ) = A ( q )[ − f ( q , τ )] + B ( q ) . To extract information from the DICF, aparameterised ISF must be fitted. For non-interacting Brownianparticles with diameter, d , f ( ~ q , τ ) = f D ( | ~ q | = q , τ ) = e − Dq τ , D = k B T / ( πη s d ) . (4)with k B T the thermal energy, η s the solvent viscosity and D theextracted diffusivity. However, flow brings anisotropy in particle PSfr repl cements − − q x [µm −1 ] − − q y [ µ m − ] FlowStandard DDM − − q x [µm −1 ] Drift-corrected − − τ [s]12 g ( n ) ⊥ ( τ ) × Sector: ⊥ n ⊥ − − τ [s] ¯g ( n ) ⊥ ( τ ) (a) (b)(c) (d) Fig. 1
Impact of shear and drift correction on DICF (a) DICF for DDMcorrelator, g ( ~ q ) (Eq. 1) at delay time τ = .
02 s , for Poiseuille flow at µ lmin − and µ m imaging depth ( h v i = µ ms − ). Colour map:light, high g values, and dark, low g values, flow direction indicated byarrow. Perpendicular ( ⊥ , cross-hatched) and near-perpendicular (n ⊥ ,hatched) sectors used to define g ( n ) ⊥ ( q ) with half-width θ = ° . (b) DICFafter drift correction, ¯ g (Eq. 8), colour scale unchanged. (c) Time depen-dence of non-corrected DICF in (a) at q = µ m − , g ( τ ) . Symbols: dark(blue), ⊥ ; and, light (grey), n ⊥ . Line, anisotropic-DDM, diffusive fit of g ⊥ ( D = . µ m / s ). (d) Drift-corrected DICF from (b): symbols, as in(c); lines, flow-DDM, Eq. 9 ( D = . µ m / s , ∆ v = µ ms − ). displacement and complexity to the ISF: to size particles we mustdisentangle microscopic dynamics and macroscopic flow. Under flowing conditions, the total displacement of a Brownianparticle, δ ~ r , is the sum of diffusive motion and ballistic motiondue to flow, δ ~ r v . Using Eq. 3, the ISF can be expressed as a prod-uct of separate processes: f ( ~ q , τ ) = ∏ i f i ( ~ q , τ ) = f D · f v · f FS , (5)which includes contributions from diffusive motion ( f D ), flow re-lated motion ( f v ) and finite size effects ( f FS ). As the total ISFis a product , whenever a single component f i → , the total ISF f → . Therefore, the fastest decorrelation process will dominatethe entire response, leaving slower processes immeasurable. Thismeans that, to measure particle size diffusion must be the fastestdecorrelation process and that we must then understand the de-tailed impact of flow on the ISF.A uniform steady flow, with velocity ~ v , will shift the position ofeach particle by δ ~ r v = ~ v τ in addition to diffusive motion, introduc-ing a phase shift into the ISF: f v ( ~ q , τ ) = e i ~ q · ~ v τ , thus ℜ { f v ( ~ q , τ ) } = cos ( ~ q · ~ v τ ) . (6)This is apparent in the DICF as ‘waves’ in the direction of flow, asillustrated in Fig. 1(a), which shows a typical experimental DICF,in the ( q x , q y ) plane at one delay time τ = .
02 s , obtained for rownian particles flowing with mean velocity h v i = µ ms − (see Sec. 3.1 for experimental details). Equation 6 implies thatflow should not contribute to the DICF in the direction perpen-dicular to the flow, g ⊥ , as ~ q · ~ v = , and f ⊥ v = . However, asimages are composed of finite-sized pixels, measurements of g ⊥ require averaging over a finite-size sector with half-width θ andthus ~ q is only approximately perpendicular, Fig. 1(a) (hatched).In practice, we found a minimum of θ ≈ ° is required to obtainmeasurable g ⊥ from a 256 pixel image. Therefore, this sectorstill contains a velocity component ( ∼ θ | ~ v | ), which introducesa decorrelation timescale ( t v ∼ / q θ | ~ v | ), that for even moderateflow velocities can dominate over diffusion ( t v ≪ t D = / Dq ).This velocity component leads to a non-monotonic (and assuredlynon-diffusive) g ⊥ ( ~ q , τ ) set by f v rather than f D , Fig. 1(c) [(blue)squares]. The non-monotonic behaviour is exacerbated in the ad-jacent sector [(grey) circles]. We refer to a simple diffusive fit to g ⊥ as “anisotropic-DDM”, a technique which has been used forparticles influenced by a magnetic field. A combination of a finite field of view and flow will also causedecorrelation due to particles leaving the image (and being re-placed by on average uncorrelated particles). This introducesa finite-size term into the total ISF, which for flow along the x direction takes the form f FS = max [( − | v x | τ / L x ) , ] , (7)where L x is the image size in the flow direction. This sets a hardupper limit for DDM-based measurements under flow, as the dif-fusive dynamics must lead to decorrelation on a timescale fasterthan L x / v x , whereupon particles disappear from the field of view. As stated, anisotropic-DDM is quickly overwhelmed by the pres-ence of flow and the remaining velocity component. Here, wepresent a new analysis scheme, flow-DDM, that allows a reductionof the flow contribution and a decoupling of the diffusive motionof Brownian particles from the background flow. Conceptually,the effect of flow on a system moving with a well defined uni-form speed, ~ v , can be minimized by simply observing its dynamicsin a co-moving frame of reference. Recording movies directly ina co-moving frame of reference is obviously challenging, but bydetermining the mean drift speed h ~ v i in the laboratory frame ofreference it is then straightforward to shift the images when com-puting the DICF. The resulting “drift-corrected DICF” can then befitted with an appropriate model that takes into account diffusivemotion and the fact that in most practical scenarios there will bea spread in flow speeds. We first need to measure the mean flow velocity, for which sev-eral methods exist such as particle tracking velocimetry or particleimaging velocimetry. However, the recently introduced methodof phase dynamic microscopy ( ϕ DM) is particularly suitable inthe current context because it is a digital Fourier method that doesnot require particle resolution and can be readily integrated withDDM. At high flow speeds the dominant change between frames
PSfr repl cements h v i [µm s −1 ]10 − t i Low q ←−− t max O ( t f ) ←−−− t D t ∆v t FS h v i [µm s −1 ]High q ↓ t D ∝ / Dq t ∆v ∝ ∆vq ↑↓ t FS ∝ L x / v (a) (b) Fig. 2
Decorrelation time ‘phase diagram’. (a) ISF decorrelationtimescales, t i , as a function of average mean drift velocity, h v i , atlow wavevector, q = µ m − . Lines: blue, diffusion time ( t D = Dq , for D = µ m / s , solid when measurable, dashed otherwise); orange (dot-dashed) velocity distribution [ t ∆ v = . / ( ∆ vq θ ) , ∆ v = . h v i ]; dark grey(solid), finite-size effect ( t FS = . L x / h v i , L x = µ m ); and dotted lines,standard DDM limits [lower, frame time limit ( ∼ t f ); and upper, movielength ( t max )]. Shading: light green, diffusion measurable; grey, decor-relation before diffusion. (b) Equivalent decorrelation time diagram athigh q = µ m − , sharing t i axis. is the translation, which in Fourier space leads to a cumulativephase shift ϕ ( ~ q ) = ~ q · ~ v τ (from Eq. 6). The drift velocity ~ v canthen be estimated from the gradient of ϕ ; by averaging over asufficiently long movie segment. The method has been shown towork over a wide range of speeds, even when the displacementsdue to random motion start to dominate. Having measured the mean flow velocity, h ~ v i , we can then com-pute the drift-corrected DICF: ¯ g ( ~ q , τ ) = D | ˜ I ( ~ q , t + τ ) e − i ~ q ·h ~ v i τ − ˜ I ( ~ q , t ) | E t = A ( ~ q )[ − ¯ f ( ~ q , τ )] + B ( ~ q ) . (8)Equation 8 allows reduction of the flow contribution, as both the’waves’ and non-monotonic behaviour of the DICF [Fig. 1(a) and(c)] are not apparent in the drift-corrected DICF [(b) and (d)].However, we note that the drift-corrected DICF is clearly not radi-ally symmetric, indicating that there is still some residual contri-bution from the flow. This is due to the fact that there is actuallya distribution of flow speeds about the mean. This speed distri-bution must be considered to allow accurate measurements ofparticle size at high flow speeds. To account for the residual effects of flow the drift-corrected ISFremains a product of three contributions, ¯ f ( ~ q , τ ) = ∏ i f i ( ~ q , τ ) = f D · f ∆ v · f FS , (9)but now including f ∆ v to account for the residual velocity distribu-tion. Such distributions in the flow velocity originate from severalcauses. Indeed, as the sample will be flowing through a geome-try with fixed boundaries, there must be a velocity gradient (orshear). As we image a finite volume due to the depth of field,this causes a range of particle speeds to be captured. Additionally,there is often a small spatial variation of velocities across the fieldof view, for example, due to the speed profile in Poiseuille flow, r the flow speed may even vary over time. In all these cases, af-ter correcting for the average velocity there will be a distributionof residual velocities, P ( ∆ ~ v ′ ) , which we characterise by the width ∆ v . In the following we assume that this residual motion is purelyin the direction of the original flow and we now drop vector no-tation for velocities (see ESI † Section S1 for comments on moregeneral residual motion).Note that the width of this distribution is in principle not setby the mean speed alone, e.g. in rheometric cone-plate flow theshear rate is fixed (fixing the velocity distribution for a given op-tical section) but the speed varies with height. However, it is im-portant to realise that in practice for a given imaging region thevelocity distribution will increase with the mean speed in a linearfashion, ∆ v = k h v i , with the proportionally constant dependent onthe flow geometry, but assumed to be less than 1 (for imagingaway from the boundaries).To size particles, we first restrict our analysis to the perpendic-ular sector, f ⊥ ∆ v , for which the impact of ∆ v is minimised (just aswith h v i for f ⊥ v ). This attempts to ensure that diffusion causesdecorrelation in Eq. 9. For tractability, we assume a uniform dis-tribution of residual velocities (- ∆ v to + ∆ v ) and use a small angleapproximation for the phase shift, i.e. ~ q · ∆ ~ v ′ τ ≈ q ∆ v ′ θ ′ τ . Integra-tion over the residual velocity distribution ( ∆ v ′ ) and then sectorangle ( θ ′ ) thus yields: f ⊥ ∆ v ( q , τ ) ∝ Z + θ − θ Z + ∞ − ∞ P ( ∆ v ′ ) exp ( − iq θ ′ ∆ v ′ τ ) d ∆ v ′ d θ ′ = Si ( q ∆ v τθ ) / ( q ∆ v τθ ) , (10)where Si is the sine integral and proportionality such that f ( τ → ) = . Note that although the assumption of a uniform P ( ∆ v ′ ) is evidently an idealisation, it is sufficient to capture the featuresof more realistic distributions within the frame work of our flow-DDM protocol (see ESI † Section S1).
To see how best to extract an accurate particle size from ¯ g ⊥ over the greatest possible range of flow speeds we must considerrelative decorrelation times for different components of the ISF,where f i ( t i ) = / e in Eq. 9. The decorrelation time for diffusion, t D = / Dq , does not depend on h v i , Fig. 2 [(blue) solid line], butit does decrease strongly with increasing q .Finite-size effects by contrast lead to t FS = . L x / v x , indepen-dent of q . Therefore decorrelation is predominantly due to dif-fusion for speeds up to v x ≈ . q DL x , i.e. this effect becomesless important at higher q , cf. Fig. 2(a) blue and dark grey lines.By acquiring images with a large field of view L x and high spatialresolution (to access high q ) finite size effects can be be greatly re-duced. But the faster dynamics at higher q also require high framerates, which in modern scientific cameras and confocal laser scan-ning systems decreases with the height L y of the image. In prac-tice, these requirements are most effectively achieved by taking arectangular image, with the long axis of the field of view alignedwith the flow direction: we use L x = L y throughout.The decorrelation time caused by the distribution of speeds, from Eq. 10, t ∆ v = . q θ∆ v (11)decreases with both the width of the speed distribution (and thusflow speed) and with q . Therefore, we can again increase the im-pact of diffusion, this time relative to ∆ v , by looking at higher q ,[ cf. (orange) dot-dashed lines, Fig. 2(a) and (b), where we take ∆ v = . h v i ], and hence measure particle size at higher speeds.Experimentally, we access high q using relatively high magnifica-tions. This has the added benefit of reducing the imaged width,and therefore the contribution to ∆ v from velocity variation in the y direction. For some microscopy methods, e.g. , brightfield, thedepth of field is also decreased at high q , reducing any contribu-tion to ∆ v from the velocity gradient in z . However, the maximumuseful magnification is limited by the drop in signal amplitude[ A ( q ) ≪ B ( q ) ], as without accessing higher q greater magnifica-tion only increases finite size effects. While we have now maximised the impact of diffusion relativeto the flow on decorrelation, we must also discriminate betweenthe two processes to determine the reliability of the measurement.For finite-size effects we can estimate t FS independently from h v i ;but, in the perpendicular sector there is no robust way to discrim-inate between ∆ v and D over a limited q range, as both f ∆ v and f D decrease monotonically. However, diffusion is isotropic, while theimpact of shear depends on angle. We therefore consider a sectorthat is adjacent to the perpendicular sector, ¯ g n ⊥ [Fig. 1(a)], with f n ⊥ ∆ v ( q , τ ) ∝ Z θθ Z + ∞ − ∞ P ( ∆ v ′ ) exp ( − iq θ ′ ∆ v ′ τ ) d ∆ v ′ d θ ′ = [ Si ( q ∆ v τθ ) − Si ( q ∆ v τθ )] / ( q ∆ v τθ ) . (12)Decorrelation due to ∆ v now occurs at a more rapid rate ( ∼ × compared to f ⊥ ∆ v ) and we can separately probe ∆ v by simultane-ously fitting two sectors of the DICF and establish whether themeasured particle size is reliable, i.e. t D ≪ { t FS , t ∆ v } . This com-bination of drift correction, imaging optimisation and fitting to-gether we term “flow-DDM”. We now turn to look at applying flow-DDM to measure particlesize for a dilute colloidal suspension and demonstrate it usingtwo different microscopy techniques and flow geometries.
First, we use a dilute ( φ = . ) suspension of sphericalpolystyrene particles in water, with a nominal diameter of
300 nm flowing through a square capillary with a controlled flowrate, giving Poiseuille flow, Fig. 3(a). Images were taken usingphase contrast microscopy (20x/0.5 objective at 400 frames persecond for t max =
20 s ). The rectangular images ( × pix-els image, . µ m /px → µ m by µ m ) are aligned along thecentre of the capillary. At a given flow rate, Q , the capillary isthen imaged at multiple focal depths, z .To establish a reference diffusion coefficient, i.e. the free- a) Capillary (cid:1) ow, Q (b) Rheometric (cid:0) ow, shear rate (cid:2) xyz ≈ Imagedregion .h L x =4L y xyz Fig. 3
Imaging and flow geometries. (a) Capillary flow, squarewith flow rate Q . Arrows indicate velocity direction and spatial variation;example imaging region shown in orange at height z with flow direction x .(b) Rheometric flow with confocal microscopy. Flow (arrows) generatedby rotating cone (grey) above glass coverslip with shear rate, ˙ γ (velocitygradient, d v /d z ). Imaging region highlighted with 4:1 aspect ratio ( L x : L y )aligned with flow direction. diffusion coefficient ( D ), quiescent samples were recorded inthe same conditions. Using standard DDM (Eqs. 1 and 4), a q -dependent diffusion coefficient was extracted, Fig. 4(a). The dif-fusivity, D = h D ( q ) i = . ( ) µ m / s (averaging over q = . to . µ m − ) implies a particle size of d = ( ) nm at ° C . To explore the general application of flow-DDM to other mi-croscopy techniques and flow geometries, we use a confocal mi-croscope coupled to a rotational stress-controlled rheometer (Anton Paar MCR 301), Fig. 3(b). Images were taken using an in-verted confocal laser-scanning microscope [Leica SP8, 20x/0.75objective)], a technique previously used with DDM to measuredense quiescent systems. The sample is a dilute ( φ = . )suspension of fluorescently-dyed poly(methyl methacrylate) par-ticles stabilised with poly(vinyl pyrrolidone); the particles are sus-pended in a density matched 21 wt.% caesium chloride solutionto prevent sedimentation and screen electrostatic interactions.Images are taken at 50 frames per second for t max =
200 s witha × resolution and . µ m pixel size ( µ m × µ m field of view).In the quiescent state, a plateau in D ( q ) is seen for q = to . µ m − , Fig. 4(b) (open circles). Due to the small imagewidth ( L y ) used for flow-DDM, “spectral leakage” leads to anapparent drop in diffusivity: at high q values, corresponding tolength scales smaller than the particle, g ( q , τ ) is distorted due toparticles cut off at the image boundaries. This effect is miti-gated by smoothing the image boundaries using a Hanning win-dow, ( cf. open and filled symbols); all further diffusion measure-ments presented are from windowed images. Averaging D ( q ) from . µ m − to . µ m − , gives D = . ( ) µ m / s and a par-ticle diameter of . ( ) µ m at ° C . At low q there is an appar-ent rise in D due to diffusion out of the optical section. Addi-tionally, we can also estimate the particle size from A ( q ) , Fig. 4.Considering high resolution fluorescence imaging of a dilute sus-pension, we expect A ( q ) to be proportional to the particle formfactor, for which a first minimum should occur at qd / ≈ . byconsidering the Fourier transform of a uniform intensity and ne-glecting the point spread function. The minimum at q = . µ m − , PSfr repl cements D [ µ m / s ] d = 298nm 0.00.10.20.30.4 d = 2.65µmNo windowWindowed0 1 2 3 4 5 q [µm − ] A , B [ A . U . ] Rest Flow AB AB 0 1 2 3 4 5 6 q [µm − ] (a) (b)(c) (d) Fig. 4
Diffusion measurements of quiescent samples. (a) Phase con-trast microscopy of a dilute suspension of
300 nm polystyrene particlesas a function of wavevector, q . (b) Confocal microscopy of a dilute col-loidal suspension, poly(methyl methacrylate) in CsCl solution, ∼ µ m .Symbols, D ( q ) for: filled (blue) squares, standard DDM protocol; and,open squares, Hanning-windowed data. (c) Signal [filled, A ( q ) ] and noise[open, B ( q ) ] for D measurements in (a). Large symbols at rest, smallsymbols under flow at h v i = µ ms − . (d) A ( q ) and B ( q ) for D in (b). Fig. 4(d), results in an estimated diameter d ≈ . µ m , in quanti-tative agreement with results from the measured D .To create flow, a °,
50 mm diameter cone-plate geometry gen-erates a uniform shear rate, ˙ γ , with the velocity gradient perpen-dicular to the imaging plane. The shear rate is set by the rota-tional speed of the rheometer. Imaging at an increasing depth intothe sample, h = and µ m , increases the translational speed h v i = ˙ γ h ; greater depths could not be used due to high sampleturbidity. Images are taken at a radius of ≈
20 mm from center ofthe cone, to ensure the direction of the rotational flow does notvary significantly along the flow direction, x . We now establish the effectiveness of flow-DDM and investigatethe limiting factors for reliable measurement. We measured dif-fusivity of a dilute colloidal suspension with increasing flow ratethrough a capillary, which we compare to the free-diffusion coef-ficient D obtained from quiescent conditions. However, the flowvelocity in a capillary varies strongly with position. We show inFig. 5 the average flow velocity h v i , measured in the ( x , y ) -planecenter of the capillary using ϕ DM and normalised to the flow rate Q , as a function of the height of the focal plane ( z ) for several Q values. We find a near parabolic flow profile, with the velocityreaching a maximum in the centre of the capillary, Fig. 5 (sym-bols), matching the velocity predicted by Boussinesq (dashedline). Temporal fluctuations in the flow speed may occur andwould be included in error bars, but no systematic variation over ∼ t max was observed. Near the centre of the channel ( z = µ m to µ m ), h v i is near constant and we therefore average overthese four positions, although we present results across the fulldepth in Fig. S2, ESI † . These measurements are away from thetop and bottom of the channel, where the strong gradient in Sfr repl cements z [µm]05001000150020002500 h v i / Q [ µ m µ l − ] x [µm]0160 y [ µ m ] Q [µl min −1 ]
310 30 332 343 v [µm s − ] Fig. 5
Velocity variation in a capillary. (a) Average drift velocity, h v i , as afunction of imaging depth, z , collapsed by flow rate, Q . Symbols: varying Q , see inset legend, error bars indicate standard deviation in v ( t ) from 100frame ( .
25 s ) subsections; dashed line, expected h v i from .
98 mm squarecapillary flow profile, averaged over µ m image width. Grey shading,positions used for particle sizing measurements. Inset: spatial variationof velocity in the centre of the channel at Q = µ lmin − , shade (color)indicates velocity (see scale above) and arrows direction. Average driftvelocity is extracted from µ m sub-regions and linearly interpolated. h v i may combine with the optical section to produce a large ∆ v .As with z , there is also a velocity variation across the channelwidth, y . Measuring h v i in sub-regions of the image we can esti-mate this variation at ≈ , with a µ ms − spatial variation for h v i = µ ms − , Fig. 5(inset).From the measured h v i , we computed the drift-corrected DICFfor all positions (Fig. S3 for typical ¯ g , ESI † ). To extract a diffu-sion coefficient we simultaneously fit the perpendicular and near-perpendicular sectors of the drift-corrected DICF, Fig. 1(b), us-ing Eqs. 4, 8–12 over a q range of . µ m − to . µ m − where A / B > . . h v i is taken as an input parameter, and { D ( q ) , ∆ v , A ( n ) ⊥ ( q ) and B ( n ) ⊥ ( q ) } as the fitting parameters (Fig. S3 for typi-cal results as a function of q , ESI † ).We varied the flow rate in the range Q = µ lmin − to µ lmin − , resulting in nearly two decades of measured h v i inthe middle region of the capillary (from µ ms − to µ ms − ).Plotting the extracted diffusivity h D i against h v i , Fig. 6(a), wefind that h D i closely matches the quiescent measurement, D , upto µ ms − , cf. filled squares and dashed line. Correspond-ingly, at the minimum q used for averaging the diffusion timescale t D is far smaller than t ∆ v and t FS , Fig. 6(b), giving great con-fidence in the accuracy of the overall analysis, as discussed inSec. 2.3.3. However, t D and t ∆ v become comparable at higher ve-locity h v i = µ ms − and so the error in h D i increases, before h D i itself increases at yet higher speeds. For sizing, this wouldappear as a smaller particle. Based on Fig. 6, we conclude thatdue to the present optimal imaging conditions ∆ v is the limitingfactor (as t ∆ v < t FS ) and that t D / t ∆ v / is necessary for reliable siz-ing measurements [Fig. 6(b) hatched region]. Using Eq. 11, thisallows us to estimate the maximum velocity, v max = µ ms − ,for reliable particle sizing by considering our measured ∆ v ≈ . v (Fig. S4, ESI † ) and θ = °. Using larger θ = . ° sectors means ∆ v will have a larger impact ( v max = µ ms − ), and correspondinglywe see a larger h D i measurement at a lower h v i . µ ms − PSfr repl cements h D i [ µ m / s ] Flow-DDM Anisotropic-DDM h v i [µm s −1 ]10 − t i [ s ] t D t ∆v t FS (a)(b) Fig. 6
Measuring diffusion with varying capillary flow rate. (a) Ex-tracted diffusivities vs mean drift velocity, h v i , averaging over 4 positionsin channel centre. Symbols: filled (blue) squares, flow-DDM averaging D ( q ) over q = . µ m − to . µ m − with θ = ° and open (black) squares,anisotropic-DDM. (b) Timescale phase diagram. Symbols, timescalesat minimum q used for flow-DDM, q = . µ m − : (blue) squares, mea-sured diffusion; solid (orange) triangles, extracted velocity distributionfrom flow-DDM; open (orange) triangles, velocity distribution from v ( t ) ,Fig. 5, using .
25 s subsections and the difference between 5 th and 95 th percentiles; and (grey) circles, finite-size effect from h v i . Lines and shad-ing scheme as in Fig. 2, with striped shading indicating factor threetimescale separation. (Fig. S5, ESI † ).A ∆ v ∼ . h v i is larger than expected from variation acrossthe width of the channel, Fig. 5(inset). It is instead related totemporal fluctuations, with ∆ v measured with flow-DDM closelymatched by the variation in v ( t ) , cf. Fig. 6(b) open and filled tri-angles. The spatio-temporal velocity fluctuations mean that thecontribution to ∆ v from the optical section is insignificant, whichresults in consistent diffusivity measurements across the capillary,even as the velocity variation across the depth of field changes,see ESI † Section 2. However, even if these measurements werenot limited by flow stability, t FS would soon impact measurements[Fig. 6(b), solid dark (grey) line], even with the rectangular fieldof view.Comparing flow-DDM to existing DDM-based techniques, wesee a significant improvement over anisotropic-DDM, i.e. using aperpendicular sector of θ = ° and a simple diffusive fit (Eq. 4)over the same q range, Fig. 6(a). Flow-DDM enables reliable mea-surement of the free diffusion coefficient, D , and thus the parti-cle size to h v i an order of magnitude faster than for anisotropic-DDM, for which h D i starts to significantly increase from h v i . µ ms − . The O ( ) × improvement is consistent with ∆ v ∼ . h v i as the particle velocities are reduced 10 fold thanks to thedrift-correction (Eq. 8).Additionally, another recent technique based on DDM but us-ing a higher-order “far-field” correlator has been suggested toeliminate the impact of translation due to flow ( i.e. h v i ). Thisfar-field correlator can be related to the magnitude of the ISF,which should be translation invariant. However, we find thateven in quiescent conditions that interpretation of this correlator
Sfr repl cements ˙ γ h [µm s −1 ]10 h v i [ µ m s − ] t [s] 1.01.2 v ( t ) / h v i ˙ γ [s −1 ] Fig. 7
Rheo-confocal flow velocity. (a) Extracted average drift velocity, h v i , as a function of applied shear rate, ˙ γ . Symbols, time averageddrift velocity extracted from phase shift between successive frames (errorbars, standard deviation in v extracted from subsections of movie;squares, imaging depth, h = µ m ; and circles, h = µ m . Shear rategiven by colour (or shade), see inset legend. Line, equality between h v i and nominal velocity, ˙ γ h . Inset: time-dependent drift velocity v ( t ) for h = µ m at ˙ γ ≥ − . is challenging, as it yields a measured D ( q ) lower than the ex-pected D (Fig. S6, ESI † ), while for flowing samples the resultsvary proportionally with non–drift-corrected DDM (Fig. S5, ESI † ).For quantitative results, we therefore use flow-DDM. We now demonstrate the general applicability of flow-DDM byusing a setup with a different microscopy method, flow geome-try and particle size. Here, we performed rheo-confocal imag-ing of micron-sized particles, and varied the flow velocity, h v i ,through the imaging height, h = or µ m , and applied shearrate, ˙ γ = .
05 s − to
10 s − . This setup allows us control of themean speed independent of the velocity spread by imaging theflow within a well-defined optical section.Figure 7 shows h v i measured from ϕ DM (symbols) as a functionof ˙ γ h . The extracted average velocity closely matches the speedpredicted for a shear flow, h v i = ˙ γ h (line). However, at high shearrates ( ˙ γ ≥ ) there are noticeable oscillations in the flow speed(see inset), consistent with a slight geometry misalignment. The drift-corrected DICF, ¯ g (Eq. 8), was therefore calculated usinga time-dependent drift velocity based upon a smoothed averageof h v i from subsections, v ( t ) . We then fit ¯ g using the protocoldeveloped for Poiseuille flow in Sec. 4.1, but now using a q rangeof . µ m − to . µ m − so that A / B remains & . . The lower q range consequently requires an increased θ of . ° to ensure anaverage over sufficient ~ q . Typical results for ¯ g and fits thereof areshown in Fig. S3, ESI † .Figure 8(a) shows the measured h D i as a function of shear rate[light (blue) symbols]. At ˙ γ / − , h D i ≈ . µ m / s , giving aninferred particle diameter of d = . µ m . The diffusivity is compa-rable to the rest measurement, D = . µ m / s , although thereis an ≈ increase that may arise from a small change in thesolvent viscosity due to temperature.In order to understand the limits of flow-DDM we againneed to compare the extracted decorrelation timescales shown in PSfr repl cements h D i [ µ m / s ] ˙ γ h h D i [ µ m / s ] Anisotropic-DDMFlow-DDM − ˙ γ [s −1 ]10 − t i [ s ] t FS t ∆v t D (a)(b) Fig. 8
Measuring diffusion in rheometric flow. (a) Diffusion coeffi-cient, D , as a function of applied shear rate, ˙ γ , [ h = µ m (squares) and h = µ m (circles)] averaging over q = . µ m − to . µ m − . Symbols:dark (blue), D from flow-DDM; and, light (grey) anisotropic-DDM. Inset:symbols, anisotropic-DDM vs nominal velocity ( ˙ γ h ). (b) Decorrelationtimes, t i at q = . µ m − for given terms from flow-DDM, symbols: small, t D ; large, t ∆ v ; and filled, t FS . Shading as in Fig. 6(b). Fig. 8(b). First we should note that the decorrelation time asso-ciated with the spread in velocities, t ∆ v , decreases with shear raterather than the velocity: t ∆ v is the same for the two heights, h , pre-sented here. This experimental data implies that ∆ v = ∆ h · ˙ γ , wherewe find ∆ h = µ m (see ESI † , Section S4 for details). This length-scale, ∆ h , is comparable to the quoted optical section of . µ m forour confocal imaging configuration, which suggests that ∆ v arisesfrom the velocity gradient across the depth of field in this shearflow. However, we cannot rule out a contribution from the time-dependent velocity as rapid changes may not be captured by thesmooth interpolation of v ( t ) . Our optimised imaging settings en-sured that finite size effects remain negligible, with t FS the slow-est of the three decorrelation processes, even at h = µ m . So,diffusion (or size) measurements are limited by the increasing ve-locity distribution, with flow-DDM again producing reliable mea-surements for t D . t ∆ v / , just as in Sec. 4.1.Using anisotropic-DDM, i.e. without drift correction, h D i showsan increase at much lower shear rates, Fig. 8(a) (black symbols),and already increases at ˙ γ & . − for h = µ m (circles). Herethe rise in h D i occurs with the flow speed (see inset) rather thanshear rate. The relative improvement seen for flow-DDM then de-pends on h , as the relevant velocity scale is changed from beingset by the imaging depth ( h v i = ˙ γ h ) to being controlled by theeffective optical section ( ∆ v = ˙ γ∆ h ): flow-DDM makes ( h / ∆ h ) × higher mean speeds accessible for size measurements. Mean-while, the far-field correlator again significantly underestimatesdiffusivity in quiescent conditions (Fig. S6, ESI † ). Thus, flow-DDM appears as an exciting new technique to accurately measurefree-diffusion and thus size particles under general flow condi-tions. Conclusions
In summary, we have proposed flow-DDM as a novel method toaccurately measure free-diffusivity, and from this determine parti-cle size, using microscopy videos of dilute suspensions of colloidalparticles under flow. We have presented its theoretical frameworkand practical implementation for optimal measurements.Flow-DDM is based on two main steps: 1) computing the drift-corrected DICF, ¯ g , from microscopy videos, which reduces the im-pact of flow onto the resulting experimental signal; and 2) fitting ¯ g using an appropriate model of the particle motion (includingdiffusion, residual flow velocities and finite-size effects) coupledwith an optimised fitting protocol that allows decoupling of theresidual flow velocity distribution from the diffusive motion. Wehave validated flow-DDM using two different particle suspensions,demonstrating its general application by studying two setups withdistinct optical imaging configuration and flow geometry: phase-contrast imaging with Poiseuille flow and confocal microscopywith rheometric flow.By performing systematic experiments as a function of flow rateand position within the sample, we have investigated the reliabil-ity and limits of flow-DDM, established its success over a largerange of flow speeds and determined how to optimise imagingparameters. In particular, we have shown that under optimisedconditions it is no longer the mean flow speed h v i but the width ∆ v of its distribution that limits the reliability of the technique. There-fore, ∆ v should be minimised by imaging away from regions witha large velocity gradient and by ensuring a steady flow. We haveidentified an empirical criterion to ensure reliable measurementsbased on the measured timescales of diffusion and residual veloc-ity, t D / t ∆ v / , which allows estimation of the maximum accessiblevelocity for reliable measurements, v max (assuming ∆ v = k h v i ). Itis important to note that v max depends on the particle size; so,based on the measured t D and t ∆ v obtained from flow-DDM, theabove criterion can also be used to give confidence to the userwhen performing flow-DDM measurements of suspensions withunknown particle-size.Using the advantages of DDM seen in quiescent systems, flow-DDM allows particle sizing in flowing samples without user in-puts or resolution of individual particles (as required for particletracking), and without the requirement of highly dilute samples(as for DLS). This extends sample possibilities for particle sizingunder flow, enabling high-throughput microfluidic testing in de-velopment or in-line testing during manufacturing of particulatesuspensions, which are so ubiquitous in industry. Moreover, weexpect the general framework of flow-DDM to be applicable toother imaging methods, such as bright-field, light-sheet, epi-fluorescence, and dark-field microscopy. Flow-DDM outperforms current digital Fourier techniques, suchas a diffusive fit of anisotropic-DDM or far-field dynamic mi-croscopy. Indeed, flow-DDM allows quantitative measurementswithin ≈ of the free-diffusion coefficient at flow speeds up toone order of magnitude faster than for anisotropic-DDM. Flow-DDM has been designed to be insensitive to the details of theflow, providing some robustness against some spatio-temporalvariations. Nevertheless, the method returns measurements of the mean flow velocity and an estimate for the residual veloc-ity spread, which characterises the combination of flow geometryand imaging properties.Finally, although we have focused entirely on probing diffusivedynamics of dilute suspensions to measure particle size, flow-DDM could also be applied to measure the collective dynamicsof dense (and relatively turbid) colloidal suspensions under flow.For example, ready measurement of microscopic particle rear-rangements alongside the bulk rheology could bring new insightsinto the understanding of non-Newtonian fluids such as shear-thickening or yield-stress suspensions and jammed emul-sions. Conflicts of interest
There are no conflicts to declare.
Acknowledgements
This project has received funding from the European ResearchCouncil (ERC) under the European Union’s Horizon 2020 re-search and innovation programme (grant agreement № s 731019and 862559), European Soft Matter Infrastructure (EUSMI)and Novel Characterisation Platform for Formulation Industry(NoChaPFI). The authors thank Andrew Schofield for particlesynthesis, and Jean-Noël Tourvieille and Sophie Galinat for en-lightening discussions. All data used are available via EdinburghDataShare at https://doi.org/10.7488/ds/2987 . References
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