Decoupling the effects of shear and extensional flows on the alignment of colloidal rods
DDecoupling the effects of shear andextensional flows on the alignment of colloidalrods
Vincenzo Calabrese, ∗ Simon J. Haward, and Amy Q. Shen ∗ Okinawa Institute of Science and Technology, Onna-son, Okinawa 904-0495, Japan
E-mail: [email protected]; [email protected]
Abstract
Cellulose nanocrystals (CNC) can be considered as model colloidal rods and havepractical applications in the formation of soft materials with tailored anisotropy. Here,we employ two contrasting microfluidic devices to quantitatively elucidate the role ofshearing and extensional flows on the alignment of a dilute CNC dispersion. Character-ization of the flow field by micro-particle image velocimetry is coupled to flow-inducedbirefringence analysis to quantify the deformation rate–alignment relationship. Thedeformation rate required for CNC alignment is 4 × smaller in extension than in shear.Alignment in extension is independent of the deformation rate magnitude, but is either0 ◦ or 90 ◦ to the flow, depending on its sign. In shear flow the colloidal rods orien-tate progressively towards 0 ◦ as the deformation rate magnitude increases. Our resultsdecouple the effects of shearing and extensional kinematics at aligning colloidal rods,establishing coherent guidelines for the manufacture of structured soft materials. a r X i v : . [ c ond - m a t . s o f t ] S e p NTRODUCTION
Colloidal rods have received long-lasting attention as building-blocks for complex materialsbecause of their effective gelling properties and their ability to orient upon hydrodynamicforces, allowing the manufacture of materials with tailored anisotropy.
Soft materials withanisotropic orientation have shown a large number of advantages, such as their mechanicalstrength, structural color, electrical and thermal conductivity, and directional controlof cell growth. To achieve directionality in soft materials, microfluidic platforms have been extensivelyused to control hydrodynamic forces, mainly through shearing and extensional-dominatedflows, to aid particle alignment while keeping negligible inertia effects. The relative strengthbetween diffusion and hydrodynamic forces is typically described by the Péclet number
P e = | E | /Dr , where | E | is the characteristic deformation rate, and the rotational diffu-sion coefficient for non-interacting rods, Dr , is described as Dr = 3 k b T ln( l/d eff ) πη s l , (1)where k b is the Boltzmann constant, T the temperature, η s the solvent shear viscosity, l is thelength and d eff the effective diameter of the rod which accounts for the thickness of the elec-tric double layer. For P e < the particles are dominated by Brownian fluctuations, whilstfor P e > , the particles are perturbed by the flow field. It has been shown, experimentallyand theoretically, that shearing flows enable a gradual alignment of anisotropic particlestowards the flow direction, where particles align with a preferential angle of 45 ◦ at P e (cid:39) and achieve orientation parallel to the flow direction, 0 ◦ , at P e (cid:29) . Although shear-dominated flows are relatively simple to study via conventional rheo-optics techniques, “pure”extensional flows in conditions where shear-forces are negligible are experimentally difficultto assess.
Consequently, much less is known on how extensional rates affect the orien-tation of particles in shear-free conditions. Nonetheless, it has been shown that extensional2ates enable an additional control on particle orientations, inducing, for instance, particlealignment perpendicular to the flow direction, which is not possible in shearing flows.
Of considerable importance, is the work of Corona et al., who investigated the effects ofshearing and extension-dominated flows on the alignment of a concentrated suspension ofcolloidal rods using a fluidic four-roll mill device. However, no significant differences couldbe discerned between shearing and extensional-dominated flows as excluded volume effectswere likely dominant at the high concentration tested. Recent work from Rosén et al. elucidated that for extension-dominated flows, generated in flow-focusing and convergingchannels, a reduced rotational diffusion coefficient was obtained when compared to shearingflows. Nonetheless, difficulties rising from interparticle interactions, particle flexibility, anda non uniform extensional rate within the microfluidic channel, hindered a comprehensivequantitative examination to decouple the effects of shearing and extensional dominated flows.To date, a quantitative comparison between shear and extensional rate–driven alignment ofrod-like particles, in conditions where interparticle interactions are negligible, is still missing.In this article we couple quantitative flow field measurements with state-of-the-art flow-induced birefringence analysis to elucidate the role of shear and extensional rates at aligningrod-like particles. A dilute cellulose nanocrystal (CNC) suspension consisting of negativelycharged rigid rod-like nanoparticles is used as a model system. MATERIALS AND METHODS
Test fluid
The CNC was purchased from CelluForce (Montreal, Canada) as an aqueous 5.6 wt% stockdispersion at pH 6.3. A 0.1wt% CNC dispersion was prepared by dilution of the stock disper-sion with deionised water and used without further treatment. Extensive characterization ofCNC from the same industrial producer is described by Bertsch et al. and Reid et al. hear Rheometry Steady shear rheology of the 0.1 wt% CNC dispersion was measured using a strain-controlledARES-G2 rotational shear rheometer (TA Instruments Inc.) equipped with a double gapgeometry (with an inner and outer gap of 0.81 and 1.00 mm, respectively) composed of astainless steel bob and a hard-anodized aluminum cup. The dispersion was covered with asolvent trap and measurements were performed at 25 ◦ C (controlled by an advanced Peltiersystem with temperature accuracy of ± ◦ C). A microfluidic slit rheometer (m-VROCRheoSense Inc.), equipped with an A10 pressure cell, 3 mm wide and µ m high, wasused to access the rheological response of the test fluid at high values of shear rate (upto ˙ γ ≈ × s − ). The experiment was carried out at ◦ C (controlled via an externalcirculating water bath with temperature accuracy of ± ◦ C).
Atomic force microscopy (AFM)
A 0.01 wt% CNC dispersion was drop-casted on a mica substrate and imaged using anatomic force microscope (Dimension ICON3, Bruker) in tapping mode. The distribution ofthe particle contour length, l , and diameter, d , was extrapolated by tracking 976 isolatedparticles using an open-source code, FiberApp. The value of d was obtained from the AFMheight profile. Microfluidic platforms
A shearing flow-dominated channel (SFC) and an optimized shape cross-slot extensionalrheometer (OSCER) were used to generate two-dimensional (2D) flows that, in discreteareas of the geometries, provide good approximations to purely shearing and purely ex-tensional flows, respectively (Figure 1). The SFC consists of a straight channel of fusedsilica glass with a rectangular cross section (length L = 25 mm along the x -axis, height H = 2 mm along the z -axis, and width W = 0 . mm along the y -axis, resulting in an aspect4atio α = H/W = 5 ), fabricated with a selective laser-induced etching (SLE) technique (Figure 1a). The OSCER is based on a planar cross-slot geometry with two incoming andoutgoing flows placed orthogonal to each other, as described by Haward et al. and de-picted in Figure 1b. The device has a height of H = 2 . mm along the z -axis and a channelwidth of W = 0 . mm at the inlets and outlets, yielding α = 10 . , generating a good ap-proximation of a 2D flow field which is extensional-dominated in a large region around thestagnation point, at x = y = 0 (see coordinate system in Figure 1b). The fluid elements arecompressed along the y -axis and extended along the x -axis (referred to as the compressionand elongation axes, respectively, Figure 1b).The flow inside the channels is driven by Nemesys low-pressure syringe pumps (Cetoni,GmbH) and Hamilton Gastight syringes, which infuse the liquid at the inlet and withdrawit at an equal and opposite volumetric rate, Q (m s − ), from the outlet. To ensure steadyflows, the highest flow rates were set at Reynolds numbers, Re = ρU W/η , of 15.4 and 6.4, forthe SFC and OSCER, respectively, considering a fluid density, ρ = 1000 kg m − , an averageflow velocity as U = Q/HW (m s − ) and a fluid viscosity η = 1 . mPa s as determined byrheological measurements (detailed in the following section). The flows were equilibrated forat least 5 s before all measurements and confirmed as being steady by inspection of the micro-particle image velocimetry ( µ -PIV, see next section) flow fields prior to their time-averaging.All the experiments were carried out at the ambient laboratory temperature ( ± ◦ C).
Micro-particle image Velocimetry ( µ -PIV) The flow fields in the SFC and OSCER devices were obtained using time averaged µ -PIV ofthe test fluid seeded with . µ m fluorescent particles (Fluoro-Max TM , Thermo Fisher), toa concentration of ≈ . wt%. The µ -PIV measurements were conducted using a volumeillumination system (TSI Inc., MN) installed on an inverted microscope (Nikon Eclipse Ti).Nikon PlanFluor objective lenses of 4 × and 10 × with numerical apertures of NA=0.13 and0.30 were used for the OSCER and SFC device, respectively. Each geometry was placed5igure 1: Schematic diagrams of (a) the shearing flow-dominated channel (SFC) and (b)the optimized shape cross-slot extensional rheometer (OSCER) with respective coordinatesystem and scale bar. The dashed arrows indicate the flow direction.with z -axis parallel to the light source and for all the flow rates tested, a sequence of at least100 image pairs were acquired at the midplane of the geometries ( z = 1 mm). For the SFC,images were acquired at a distance of ≈ L/ from the inlets to ensure fully developed velocityprofiles. The average displacement of the seeded particles between the two images in eachpair was kept constant at ≈ δz m , corresponding to thedepth over which the seeded particles contribute to the determination of the velocity fieldwas δz m = 150 µ m and µ m , for the OSCER and SFC, respectively. Cross-correlationbetween image pairs provided velocity vectors on a square grid with a spatial resolution of2.0 and 0.8 µ m /pixel for the OSCER and SFC, respectively. Data analysis was performedusing a custom-made Matlab routine. Flow-induced birefringence (FIB)
Flow-induced birefringence (FIB) measurements were performed using an Exicor MicroIm-ager (Hinds Instruments, Inc., OR). Monochromatic light of wavelength λ = 450 nm wasshone through a linear polarizer at 0 ◦ , a photoelastic modulator (PEM) at 45 ◦ , the SFC6r the OSCER containing the testing fluid, a PEM at 0 ◦ and a linear polarizer at 45 ◦ , inthe order given. The geometries were imaged using a 5 × objective in the same position asdescribed for µ -PIV. The instrument performs Mueller matrix decomposition, determiningthe elements of 4 × R , describing the total phase shift occurring between the two orthogonally polarized lightbeams, and the orientation of the slow optical axis (extraordinary ray), θ , were obtainedfrom a total of 7 images acquired at 1 frame/s. The retardance, R (measured in nm), wasthen converted to birefringence as ∆ n = R/H . The background value of ∆ n was determinedfor the test fluid at rest and subtracted for all the analysis presented. The background valuedetermined in both geometries was ∆ n ≈ × − and comparable to the instrument reso-lution of ∆ n ≈ × − . The spatial resolution of the measurement was ≈ µ m /pixel anddata analysis was performed using a custom-made Matlab routine. RESULTS AND DISCUSSION
Characterization of the test fluid
We begin by estimating the volume density, ν = N/V , of the 0.1 wt% CNC dispersion,where N is the number of rods and V is the sample volume. In suspensions of monodisperserod-like particles, the volume density, ν , is commonly used to distinguish between the di-lute, semi-dilute and concentrated regimes. For monodisperse rods, the calculation of ν isstraightforward as a single particle length ( l ) and diameter ( d ) are sufficient to describe thewhole particle population. Contrarily, for a polydisperse distribution of rods such as CNC,the large span of lengths must be considered for a more accurate estimation of ν . Thus, thesize distribution of the CNC was extrapolated from atomic force microscopy (AFM) images(Figure 2a,b). The average contour length was (cid:104) l (cid:105) = 260 ± nm and the average diameter7 d (cid:105) = 4 . ± . nm. The effective volume density, ν e , was thereafter estimated as ν e = N e V = (cid:80) l max i =1 (cid:18) V CNC φ i V cyl i (cid:19) V , (2)where N e is the effective number of rods, l max is the longest detected contour length (700 nm), V CNC is the volume of the CNC in the sample (which can be estimated using a density of1500 kg m − ), φ i is the volume fraction of the rods with length i and V cyl i is the volumeoccupied by a single rod with length i , for which a cylindrical morphology can be approx-imated. It is noted that the distribution of the CNC diameters, d , is not accounted for ineqn. 2 since it has only a minimal effect on the estimation of ν e of slender objects and (cid:104) d (cid:105) is used to obtain V cyl i . For the 0.1 wt% CNC, the ν e ≈ / (cid:104) l (cid:105) , indicating that at this con-centration, the CNC dispersion is at the onset of the semi-dilute regime, where the particlesrarely interact (although more frequently than in the dilute regime). The absence of pro-nounced interparticle interactions was reported by Bertsch et al. for concentrations below0.5 wt%, as shown by small angle X-ray scattering studies of CNC from the same source asthat used in the present work. In addition, from the AFM images, a persistence length, l p ≈ (cid:104) l (cid:105) was extrapolated using the method of the mean-squared midpoint displacement(MSMD) within the FiberApp routine and detailed by Usov and Mezzenga, indicating thatCNC can be well described as rigid rods. A value of l p (cid:29) l was also obtained when l p wascalculated as l p = π (cid:104) d (cid:105) G k b T , (3)using the reported values of the CNC elastic modulus, G , between 5 and 150 GPa. Thesteady shear rheology of the 0.1 wt% CNC dispersion is shown in Figure 2c. The shear stress, σ , of the 0.1 wt% CNC suspension followed a linear relationship with the shear rate, ˙ γ , as fora Newtonian fluid with shear viscosity η = 1 . mPa s. Anisotropic particles are expected toexhibit a viscosity plateau at low shear rate, namely the zero shear viscosity, η , followed by8igure 2: (a) Tapping-mode atomic force microscopy (AFM) image of a drop-casted 0.01wt%CNC dispersion on mica substrate. (b) Particle contour length, l , and particle diameter, d (inset in (b)), distributions as obtained from the AFM counting of 976 isolated particles.(c) Flow curve of the 0.1 wt% CNC dispersion presented as shear stress, σ vs shear rate, ˙ γ .The black circles are data obtained using a rotational strain controlled rheometer (ARES-G2). The data represented by the diamonds are obtained using a microfluidic slit rheometer(m-VROC). The dashed (black) and solid (blue) lines are the predictions of the zero shearviscosity from eqn.4 using values of average length, (cid:104) l (cid:105) , and maximum length, l max , respec-tively, as obtained from AFM analysis.a shear thinning behavior up to a second viscosity plateau at higher shear rates, η ∞ . From eqn. 1 it is clear that longer rods will align at lower shear rates than shorter rods since Dr ∝ /l . As such, in polydisperse suspensions of non-interacting colloidal rods, the onsetof shear thinning is dictated by the longest population of rods. Substituting l with l max andaccounting for the contribution of the electric double layer ( δd = 22 . nm) to the effective9iameter ( d eff ≈ δd + (cid:104) d (cid:105) ) in eqn. 1, we estimated the expected onset of the shear thinningat ˙ γ (cid:39) s − ( P e (cid:39) ). However, at this CNC concentration, the value of viscosity of thetest fluid is close to that of the solvent shear viscosity, η s , making practically impossible tocapture the shear thinning region expected for anisotropic particles. For a dilute dispersionof monodisperse rods, η can be estimated as η (cid:39) η s + ν e k b T (cid:18)
130 1 Dr (cid:19) . (4)When the value of l in eqn. 1 is substituted with either (cid:104) l (cid:105) or l max , the predicted values ofzero shear viscosity well encompass the experimental data (Figure 2c). The lack of anyfurther contribution needed to account interparticle interactions in eqn. 4 suggests that the0.1 wt% CNC dispersion can be considered in the dilute regime. Flow profiles
The bulk rheometry gives a good indication that the diluted CNC dispersion behaves as aNewtonian fluid. Consequently, no information regarding the onset of alignment, as usuallyassociated with the onset of shear thinning, can be discerned from the flow curve. To define adeformation rate–alignment relationship of the dilute CNC dispersion we focus on the controlof the shear rate, ˙ γ , and extensional rate, ˙ ε in two separate microchannels. The SFC was usedto provide a well approximated 2D shearing flow. Figure 3a shows two representative timeaveraged flow fields obtained by µ -PIV at the midplane ( z = 1 mm) of the SFC. The velocitymagnitude | V | , as measured by µ -PIV, is scaled by the average flow velocity | U | . The flowfield displays a velocity gradient across the channel width ( y -axis) with the greater velocityalong the centerline. This is evident when plotting the spatially averaged velocity, namely (cid:104)| V |(cid:105) (determined by averaging | V | along 0.2 mm of the x -axis) as (cid:104)| V |(cid:105) / | U | vs the channelwidth ( y -axis), where a parabolic (Poiseuille) flow profile is displayed (Figure 3b). Themeasured flow profile is in quantitative agreement with an infinite series analytical solution10igure 3: (a) Time averaged results of flow velocimetry ( µ -PIV) with superimposed stream-lines for the SFC containing the 0.1 wt% CNC dispersion. Two representative averageflow velocities, | U | , are displayed. (b) Normalized spatially averaged velocity profiles takenacross the y -axis of the channel, for values of | U | as in (a). (c) Magnitude of shear rate, | ˙ γ | = ∂ | V x | /∂ | y | , as obtained from the velocity profile displayed in (b). (d) Magnitude of theshear rate obtained from the average of | ˙ γ | at y = ± . mm, | ˙ γ | ( ± . , as a function of theaverage flow velocity. In the panels (b), (c), and (d), the solid lines are the infinite seriesanalytical solution for creeping Newtonian flow. for creeping Newtonian flow as depicted by the solid line in Figure 3, consistent with theNewtonian-like behavior described by the rheological measurement in Figure 2. From thevelocity profile displayed in Figure 3b, we compute the shear rate profile as | ˙ γ | = ∂ | V x | /∂ | y | ,where | V x | is the x -component of velocity (Figure 3c). The magnitude of the shear rate is alsoin good agreement with the expectation for a Newtonian fluid (black lines). Since the shearrate varies substantially along the channel y -axis, we select the value of the shear rate at thelocation y = ± . mm, namely | ˙ γ | ( ± . (computed as the average of the | ˙ γ | at y = 0 . mmand y = − . mm), which is the mid-point in | y | between the minimum and the maximumvalue of | ˙ γ | . The location y = ± . mm is also far from the channel side walls, where the µ -PIV limitations become evident, while still providing relatively high values of shear rate.11igure 4: (a) Time averaged results of flow velocimetry ( µ -PIV) with superimposed stream-lines for the OSCER containing the 0.1 wt% CNC dispersion. Two representative averageflow velocities, | U | , are displayed. (b) Velocity component along the elongation axis, V x ,at y = 0 mm. (c) Velocity component along the compression axis, V y , at x = 0 mm. (d)Magnitude of the extension rate, | ˙ ε | , as obtained along the compression and elongation axes.The solid line is the expected relationship for a Newtonian fluid. The relation between | ˙ γ | ( ± . and the average velocity | U | can therefore be establishedas shown in Figure 3d, leading to | ˙ γ | ( ± . = 176 | U | , which satisfies the expectation for aNewtonian fluid (solid line).The OSCER is used to generate a 2D flow field with an extensional-dominated flowin a large region around the stagnation point (the point of zero velocity at x = y = 0 ,Figure 1b). The two incoming and outgoing flows are orthogonal to each other andindicated by the dashed arrows in Figure 4a. The µ -PIV profiles for two characteristicaverage velocities, | U | , display a symmetric and Newtonian-like behavior (Figure 4a). The x -component of velocity, V x , obtained along the elongation axis (at y = 0 ) increases linearlywith x (Figure 4b) and, analogously, the y -component of velocity, V y (at x = 0 ) along the12ompression axis, decreases linearly with y (Figure 4c). This indicates uniform extensionalrates along both the compression and elongation axes. As such, the values of compressionand elongation rates can be directly obtained from the slope of the plots in Figure 4b,c. InFigure 4d, the magnitude of the compression and elongation rate as a function of the averageflow velocity | U | collapsed on a single curve, indicating that the extensional rates, ˙ ε , along thecompression and extension axes are equal and opposite, as for an ideal planar extension. The | ˙ ε | followed a linear relationship with | U | as | ˙ ε | = 1070 | U | (indicated by the solid line),in good agreement with previously reported Newtonian fluids. This behavior suggeststhat the diluted CNC dispersion behaves as a Newtonian fluid also in extension, givingan extensional viscosity η E = 4 η where η is the shear viscosity obtained from rheometrymeasurements displayed in Figure 2c and the proportionality factor of 4 is the Trouton ratiofor Newtonian fluids in planar elongation flow. Flow-induced alignment
Since the spatially resolved deformation rates, | ˙ γ | and | ˙ ε | , have been established, we shiftour attention to the impact that the two different deformation rates have on the structuralorientation of the dilute CNC dispersion. The contourplot in Figure 5a displays the FIBfields in the SFC at two different values of | U | . The birefringence intensity, ∆ n , is displayedby the contourplot and describes the extent of anisotropy in the system and the orientationof the slow optical axis, θ , is displayed by the superimposed solid segments, which directlyprobes the CNC orientation angle. As a metric of comparison, it has been recentlyshown that both ∆ n and θ yield excellent agreement with parameters describing the extent ofalignment and favourable orientation extrapolated from small angle X-ray scattering (SAXS)of anisotropic particles, enabling a trustworthy comparison of the FIB with the rheo-smallangle scattering (SAS) literature. The FIB fields in the SFC display the minimum at y = 0 (where ˙ γ = 0 s − ) and increases towards the channel walls (Figure 5a). Note that weattribute the narrow region of low FIB signal very close to the walls to “shadowing” by the13all, i.e. due to a slight imprecision of the orthogonal alignment of the microfluidic device onthe imaging system (see Figure S1 for ∆ n profiles across the channel width). Since ∆ n scaleswith the volume fraction of aligned particles ( φ aligned ), it is clear that a greater number ofCNC become aligned as the shear rate is increased. The CNC orientation, displayed by thesolid segments in Figure 5a, shows an overall alignment of the particles in the flow direction.However, for the lowest value of | U | , the CNC orientation is not perfectly mirrored betweenthe portion of the channel at y < mm and y > mm, and a degree of heterogeneity in theCNC orientation angle is observed.In the OSCER device, a large region around the stagnation point displays a strong ∆ n signal, as clearly visible for the larger value of | U | in Figure 5b (displayed by the light bluecolor in the contourplot). The direction of the CNC alignment in the compression axis isperpendicular to the flow direction ( y -axis) due to the deceleration of the fluid element uponapproaching the stagnation point and the consequent negative extensional rate along the y -axis. Contrarily, the CNC aligns parallel to the flow ( x -axis) along the elongation axis dueto the positive extension rate. Similar orientation trends have been reported for suspensionsof anisotropic particles in extensional-dominated flows (in cross slots and fluidic four-rollmill devices), although in conditions where interparticle interactions play a crucial role onparticle alignment. To have a good quantitative description of the deformation rate–alignment relationship,we plot the spatially resolved values of ∆ n as a function of the relevant deformation rate pre-viously determined in the specific locations of the SFC and OSCER geometries (Figure 5c).For the SFC, the spatially averaged birefringence signal, (cid:104) ∆ n (cid:105) , is obtained at y = ± . mmaveraging ∆ n along 1 mm in the x -direction, whilst for the OSCER device, (cid:104) ∆ n (cid:105) is obtainedby averaging ∆ n along 1 mm of the elongation axis (at y = 0 over − . ≤ x ≤ . mm, seeFigure S1, S2 for the spatially-resolved ∆ n profiles). The curves showed in Figure 5c for theSFC and the OSCER device display a similar increase in (cid:104) ∆ n (cid:105) with the deformation rate, | E | , which is well captured by a power law trend as (cid:104) ∆ n (cid:105) = A | E | p , where A is a proportion-14igure 5: Time averaged FIB profiles of a 0.1 wt% CNC dispersion, for the (a) SFC and (b)OSCER device. The birefringence, (cid:104) ∆ n (cid:105) , displayed by the contourplot whilst the directionof the slow optical axis, θ , indicated by the the solid segments (in white or black, for easyvisualization). (c) Spatially averaged birefringence, (cid:104) ∆ n (cid:105) , and (d) spatially averaged orien-tation angle, (cid:104) θ (cid:105) , as a function of the magnitude of shear rate, | ˙ γ | , and extension rate | ˙ ε | .The (cid:104) ∆ n (cid:105) and (cid:104) θ (cid:105) are obtained in the same channel location as for the relevant deformationrate. The inset in (c) displays the plot as in (c) with a re-scaled x -axis as | ˙ ε | . The solidlines in (c) are power law fittings. The dashed (black) and solid (blue) vertical lines are thevalues of Dr and Dr E (s − ) estimated via eqn.1, using values of the solvent shear viscosity( η s ), and solvent extensional viscosity ( η s,E ), respectively, and l = l max .ality factor and p is the power law exponent. For the SFC, A = (0 . ± . × − sand p = 0 . ± . , whilst for OSCER, A = (1 . ± . × − s and p = 0 . ± . .Importantly, the FIB technique enabled to capture the CNC alignment with the deforma-tion rate, otherwise unexpected by interpretation of the rheological measurements alone (seeFigure 2c). It is noted that for a given applied | U | (hence ˙ γ profile, Figure 3c), this em-pirical power law is able to capture the (cid:104) ∆ n (cid:105) profile across most of the SFC width withreasonable accuracy (see Figure S1 for spatially-resolved ∆ n profiles with the predictions15f the empirical power law fitting). Interestingly, from Figure 5c, it is apparent that lowerextensional rates are required to induce rod alignment in the OSCER device, than the shearrates required to induce alignment in the SFC. This appears to indicate that extensionalforces are more effective at inducing CNC alignment. The (cid:104) ∆ n (cid:105) profiles obtained for theextensional and shearing flow collapsed on a single master curve when scaling the exten-sional rate as | ˙ ε | (inset in Figure 5c). This indicates that extensional and shear forceshave a similar relationship with φ aligned and that the extensional forces are 4 times moreeffective for the alignment of anisotropic particles when compared to shear forces. Thereforefor a Newtonian fluid containing non-interacting rods, the following proportionality can beproposed ∆ n ∝ | ˙ γ | . ∝ | ˙ ε | . ∝ φ aligned . It is noted that the proportionality factor of 4corresponds exactly to the Trouton ratio for Newtonian fluids in a planar elongation flow,which set the relationship between the solvent shear viscosity, η s , and the solvent exten-sional viscosity, η s,E = 4 η s . The strong dependence of the solvent viscosity on the onsetof flow alignment is reflected by the Dr ∝ /η s in eqn.1. Assuming a dominant extensionalviscosity η s,E along the extensional axis, it is plausible to substitute η s with η s,E = 4 η s ineqn. 1, yielding a Dr in extension ( Dr E ) 4 times smaller compared to the Dr . Although notvalidated, the consideration of η s or η s,E for a rotational diffusion coefficient that accountsfor shearing and extensional-dominated flows, respectively, has been previously proposed byQazi et al. A decreased rotational diffusion coefficient in extensional-dominated flows hasalso been demonstrated by Rosén et al. for interacting cellulose nanofibrils. However, theabsence of interparticle interactions and the uniform, shear-less, elongational flow presentedin this work enables the first quantitative elucidation of differences between the rotationaldynamics of colloidal rods in shear and extensional flows.Considering that the longest CNC population ( l max ) aligns at lower deformation rates,the predicted onset of alignment based on Dr (dashed, black, lines in Figure 5c,d) and Dr E (solid, blue, lines in Figure 5c,d) do not exactly match the onset of birefringence. However,the scaling factor between the curves in Figure 5c strongly suggests that the larger value of16 s,E compared to η s is the cause of the earlier onset of alignment.To better evaluate the onset of alignment in shear and extensional flow, we further eval-uate quantitatively the CNC orientation as captured by the angle of orientation, θ , withrespect to the flow direction (Figure 5d). For the SFC, the spatially averaged angle of theslow optical axis, (cid:104) θ (cid:105) , was determined in the same location as for the (cid:104) ∆ n (cid:105) , i.e. averaging | θ | along 1 mm in the x -direction at y = ± . mm, whilst for the OSCER device, the (cid:104) θ (cid:105) was obtained from averaging θ along 1 mm of the elongation axis at y = 0 (averaging θ over − . ≤ x ≤ . mm) and compression axis at x = 0 (averaging θ over − . ≤ y ≤ . mm,see Figure S2 for θ profiles along the extensional axes). In shear, for | ˙ γ | < s − , withinexperimental error (note error bars), the CNC alignment is almost constant at (cid:104) θ (cid:105) (cid:39) ◦ with respect to the flow direction. For | ˙ γ | (cid:38) s − , (cid:104) θ (cid:105) progressively decreases to valuesof around ◦ at the highest shear rates, in good agreement with SAXS and birefringencestudies of a diluted CNC suspension. Along the elongation axis of the OSCER, the CNCaligns parallel to the flow, with an orientation of (cid:104) θ (cid:105) (cid:39) ◦ to the flow direction. Contrar-ily, along the compression axis, the CNC align perpendicular to the flow with (cid:104) θ (cid:105) (cid:39) ◦ .Such perpendicular alignment of elongated particles has been previously reported for shearthinning fluids in extensional-dominated flows. However, it is clear from our results thatshear thinning is not a requirement for such orientation to occur.As previously anticipated, the expected value for the onset of CNC alignment is | ˙ γ | (cid:39) s − (depicted by Dr , dashed (black) lines in Figure 5c,d) when considering the longest CNCpopulation. Although the onset of birefringence, (cid:104) ∆ n (cid:105) , occurs at values of | ˙ γ | ≈ s − ,the predicted onset of alignment at | ˙ γ | (cid:39) s − is in close agreement with the origin ofthe significant decrease in (cid:104) θ (cid:105) in the SFC (Figure 5d). The coexistence of (cid:104) ∆ n (cid:105) > andthe steady value of (cid:104) θ (cid:105) with a relatively large error at | ˙ γ | (cid:46) s − , suggests the presenceof an intermediate shear rate region where Brownian diffusion and deformation rate are ofcomparable magnitude, leading to a time-dependent and collective particle rotation in thevelocity-gradient plane. Contrarily, a Brownian dominated region occurs for | ˙ γ | < s − ,17here the system is isotropic and (cid:104) ∆ n (cid:105) ≈ , whilst, at | ˙ γ | (cid:38) s − , the hydrodynamic forcesorient the particles towards a (cid:104) θ (cid:105) = 0 ◦ , causing a pronounced increase in (cid:104) ∆ n (cid:105) . A similarintermediate shear rate regime has been described for a concentrated CNC suspension aswell as wormlike micelles and polymeric liquid crystals. For the elongational flow, (cid:104) θ (cid:105) is constant over the whole range of | ˙ ε | , in agreement withthe reported analytical solutions for diluted Brownian suspensions of rod-like particles. Therefore, the onset of a convection-dominated particle alignment in a shear-dominatedflow is more adequately described by the orientation angle (cid:104) θ (cid:105) , whilst, for the extensional-dominated flow, the onset of birefringence is a better parameter to consider since (cid:104) θ (cid:105) isindependent of the rate of deformation. When this is considered, the onset of alignmentis satisfactorily captured by a diffusion coefficient that accounts for the longest particlepopulation and the appropriate viscosity that the fluid experiences in the specific locationof the channel, i.e. Dr or Dr E . CONCLUSION
Combining flow visualization and flow induced birefringence, we are able to decouple theeffects of shear and extensional forces on a dilute dispersion of cellulose nanocrystals, con-sisting of negatively charged polydisperse and rod-like rigid particles. Simultaneous analysisof the birefringence with the orientation of the slow optical axis elucidate that extensionalforces are ca.
Acknowledgement
The authors gratefully acknowledge the support of Okinawa Institute of Science and Tech-nology Graduate University with subsidy funding from the Cabinet Office, Government ofJapan. S.J.H. and A.Q.S. also acknowledge financial support from the Japanese Societyfor the Promotion of Science (JSPS, Grant Nos. 18K03958 and 18H01135) and the JointResearch Projects (JRPs) supported by the JSPS and the Swiss National Science Foun-dation (SNSF). The authors thank Dr. Riccardo Funari from the Micro/Bio/Nanofluidicsunit at OIST for his assistance with AFM measurements and Dr. Vikram Rathee from theMicro/Bio/Nanofluidics unit at OIST for comments on the manuscript.
References (1) Solomon, M. J.; Spicer, P. T. Microstructural regimes of colloidal rod suspensions, gels,and glasses.
Soft Matter , , 1391–1400.192) Rosén, T.; Hsiao, B. S.; Söderberg, L. D. Elucidating the opportunities and challengesfor nanocellulose spinning. Advanced Materials , 2001238.(3) Calabrese, V.; da Silva, M. A.; Porcar, L.; Bryant, S. J.; Hossain, K. M. Z.; Scott, J. L.;Edler, K. J. Filler size effect in an attractive fibrillated network: a structural andrheological perspective.
Soft Matter , , 3303–3310.(4) Håkansson, K. M. O.; Fall, A. B.; Lundell, F.; Yu, S.; Krywka, C.; Roth, S. V.; San-toro, G.; Kvick, M.; Prahl Wittberg, L.; Wågberg, L.; Söderberg, L. D. Hydrodynamicalignment and assembly of nanofibrils resulting in strong cellulose filaments. NatureCommunications , , 4018.(5) Nechyporchuk, O.; Håkansson, K. M. O.; Gowda.V, K.; Lundell, F.; Hagström, B.;Köhnke, T. Continuous assembly of cellulose nanofibrils and nanocrystals into strongmacrofibers through microfluidic spinning. Advanced Materials Technologies , ,1800557.(6) Liu, D.; Wang, S.; Ma, Z.; Tian, D.; Gu, M.; Lin, F. Structure-color mechanism ofiridescent cellulose nanocrystal films. RSC Advances , , 39322–39331.(7) Kiriya, D.; Kawano, R.; Onoe, H.; Takeuchi, S. Microfluidic control of the internalmorphology in nanofiber-based macroscopic cables. Angewandte Chemie - InternationalEdition , , 7942–7947.(8) Xin, G.; Zhu, W.; Deng, Y.; Cheng, J.; Zhang, L. T.; Chung, A. J.; De, S.; Lian, J.Microfluidics-enabled orientation and microstructure control of macroscopic graphenefibres. Nature Nanotechnology , , 168–175.(9) De France, K. J.; Yager, K. G.; Chan, K. J. W.; Corbett, B.; Cranston, E. D.; Hoare, T.Injectable anisotropic nanocomposite hydrogels direct in situ growth and alignment ofmyotubes. Nano Letters , , 6487–6495.2010) Doi, M.; Edwards, S. F. Dynamics of rod-like macromolecules in concentrated solution.Part 1âĂŤBrownian motion in the equilibrium state . Journal of the Chemical Society,Faraday Transactions 2: Molecular and Chemical Physics , , 560–570.(11) Vermant, J.; Yang, H.; Fuller, G. G. Rheo-optical determination of aspect ratio andpolydispersity of nonspherical particles. AIChE Journal , , 790–798.(12) Dhont, J. K. G.; Briels, W. J. Soft Matter Vol. 2 ; Wiley-VCH: Weinheim, Germany,2007; Vol. 2; pp 216–283.(13) Reddy, N. K.; Natale, G.; Prud’homme, R. K.; Vermant, J. Rheo-optical analysis offunctionalized graphene suspensions.
Langmuir , , 7844–7851.(14) Winkler, R. G.; Mussawisade, K.; Ripoll, M.; Gompper, G. Rod-like colloids and poly-mers in shear flow: a multi-particle-collision dynamics study. Journal of Physics: Con-densed Matter , , 3941–3954.(15) Haward, S. J.; McKinley, G. H.; Shen, A. Q. Elastic instabilities in planar elongationalflow of monodisperse polymer solutions. Scientific Reports , , 33029.(16) Lang, C.; Hendricks, J.; Zhang, Z.; Reddy, N. K.; Rothstein, J. P.; Lettinga, M. P.;Vermant, J.; Clasen, C. Effects of particle stiffness on the extensional rheology of modelrod-like nanoparticle suspensions. Soft Matter , , 833–841.(17) Trebbin, M.; Steinhauser, D.; Perlich, J.; Buffet, A.; Roth, S. V.; Zimmermann, W.;Thiele, J.; Förster, S. Anisotropic particles align perpendicular to the flow direction innarrow microchannels. Proceedings of the National Academy of Sciences of the UnitedStates of America , , 6706–6711.(18) Qazi, S. J. S.; Rennie, A. R.; Tucker, I.; Penfold, J.; Grillo, I. Alignment of dispersionsof plate-like colloidal particles of Ni(OH) induced by elongational flow. Journal ofPhysical Chemistry B , , 3271–3280.2119) Pignon, F.; Magnin, A.; Piau, J.-M.; Fuller, G. G. The orientation dynamics of rigidrod suspensions under extensional flow. Journal of Rheology , , 371–388.(20) Corona, P. T.; Ruocco, N.; Weigandt, K. M.; Leal, L. G.; Helgeson, M. E. Probing flow-induced nanostructure of complex fluids in arbitrary 2D flows using a fluidic four-rollmill (FFoRM). Scientific Reports , , 15559.(21) Rosén, T.; Mittal, N.; Roth, S. V.; Zhang, P.; Lundell, F.; Söderberg, L. D. Flow fieldscontrol nanostructural organization in semiflexible networks. Soft Matter , ,5439–5449.(22) Hasegawa, H.; Horikawa, Y.; Shikata, T. Cellulose nanocrystals as a model substancefor rigid rod particle suspension rheology. Macromolecules , , 2677–2685.(23) Bertsch, P.; Isabettini, S.; Fischer, P. Ion-induced hydrogel formation and nematicordering of nanocrystalline cellulose suspensions. Biomacromolecules , , 4060–4066.(24) Bertsch, P.; Sánchez-Ferrer, A.; Bagnani, M.; Isabettini, S.; Kohlbrecher, J.;Mezzenga, R.; Fischer, P. Ion-induced formation of nanocrystalline cellulose colloidalglasses containing nematic domains. Langmuir , , 4117–4124.(25) Reid, M. S.; Villalobos, M.; Cranston, E. D. Benchmarking cellulose nanocrystals: Fromthe laboratory to industrial production. Langmuir , , 1583–1598.(26) Usov, I.; Mezzenga, R. FiberApp: An open-source software for tracking and analyzingpolymers, filaments, biomacromolecules, and fibrous objects. Macromolecules , ,1269–1280.(27) Haward, S. J.; Oliveira, M. S. N.; Alves, M. A.; McKinley, G. H. Optimized cross-slotflow geometry for microfluidic extensional rheometry. Physical Review Letters , , 128301. 2228) Burshtein, N.; Chan, S. T.; Toda-Peters, K.; Shen, A. Q.; Haward, S. J. 3D-printedglass microfluidics for fluid dynamics and rheology. Current Opinion in Colloid andInterface Science , , 1–14.(29) Haward, S. J.; Kitajima, N.; Toda-Peters, K.; Takahashi, T.; Shen, A. Q. Flow ofwormlike micellar solutions around microfluidic cylinders with high aspect ratio andlow blockage ratio. Soft Matter , , 1927–1941.(30) Meinhart, C. D.; Wereley, S.; Gray, M. Volume illumination for two-dimensional particleimage velocimetry. Measurement Science and Technology , , 809–814.(31) Wagner, R.; Raman, A.; Moon, R. J. ; 2010; pp 309–316.(32) Lang, C.; Kohlbrecher, J.; Porcar, L.; Radulescu, A.; Sellinghoff, K.; Dhont, J. K. G.;Lettinga, M. P. Microstructural understanding of the length- and stiffness-dependentshear thinning in semidilute colloidal rods. Macromolecules , , 9604–9612.(33) Usov, I.; Nyström, G.; Adamcik, J.; Handschin, S.; Schütz, C.; Fall, A.; Bergström, L.;Mezzenga, R. Understanding nanocellulose chirality and structureâĂŞproperties rela-tionship at the single fibril level. Nature Communications , , 7564.(34) Lang, C.; Lettinga, M. P. Shear flow behavior of bidisperse rodlike colloids. Macro-molecules , , 2662–2668.(35) Tanaka, R.; Saito, T.; Ishii, D.; Isogai, A. Determination of nanocellulose fibril lengthby shear viscosity measurement. Cellulose , , 1581–1589.(36) Kobayashi, H.; Yamamoto, R. Reentrant transition in the shear viscosity of dilute rigid-rod dispersions. Physical Review E , , 051404.(37) Lang, C.; Kohlbrecher, J.; Porcar, L.; Lettinga, M. The connection between biaxialorientation and shear thinning for quasi-ideal rods. Polymers , , 291.2338) Shah, R.; London, A. In Laminar Flow Forced Convection in Ducts: a Source Book forCompact Heat Exchanger Analytical Data, Academic Press, New York, 1978. ; Hart-nett, T. I. J. P., Ed.; Academic Press, 1978.(39) Haward, S. J. Microfluidic extensional rheometry using stagnation point flow.