I. INTRODUCTION

Section:

ChooseTop of pageABSTRACTI. INTRODUCTION <<II. SHEARING FLOWIII. EXTENSIONAL FLOWIV. CONCLUSIONS AND OUTLO...REFERENCESNaturally derived colloidal rods (CR) are involved in a myriad of processes and applications. Well-known examples of naturally derived CR are cellulose nanocrystals [CNCs, shown in Fig. 1(a)] and cellulose nanofibers (CNFs), both of which are extracted from wood pulp and employed in fiber spinning and biomedical applications.1,21. S. J. Eichhorn, “Cellulose nanowhiskers: Promising materials for advanced applications,” Soft Matter 7, 303–315 (2011). https://doi.org/10.1039/C0SM00142B2. K. Li, C. M. Clarkson, L. Wang, Y. Liu, M. Lamm, Z. Pang, Y. Zhou, J. Qian, M. Tajvidi, D. J. Gardner, H. Tekinalp, L. Hu, T. Li, A. J. Ragauskas, J. P. Youngblood, and S. Ozcan, “Alignment of cellulose nanofibers: Harnessing nanoscale properties to macroscale benefits,” ACS Nano 15, 3646–3673 (2021). https://doi.org/10.1021/acsnano.0c07613 Other types of carbohydrate-based CR can form via lateral aggregation of soluble carbohydrate chains in specific ionic environments, as reported for low acetylated gellan gum and carrageenan polysaccharides.3,43. M. Diener, J. Adamcik, A. Sánchez-Ferrer, F. Jaedig, L. Schefer, and R. Mezzenga, “Primary, secondary, tertiary and quaternary structure levels in linear polysaccharides: From random coil, to single helix to supramolecular assembly,” Biomacromolecules 20(4), 1731–1739 (2019). https://doi.org/10.1021/acs.biomac.9b000874. M. Diener, J. Adamcik, J. Bergfreund, S. Catalini, P. Fischer, and R. Mezzenga, “Rigid, fibrillar quaternary structures induced by divalent ions in a carboxylated linear polysaccharide,” ACS Macro Lett. 9, 115–121 (2020). https://doi.org/10.1021/acsmacrolett.9b00824 Protein-based CR, referred to as protein nanofibrils or amyloids [Fig. 1(b)], are formed via self-assembly of hydrolyzed peptides under specific conditions of flow, pH, and ionic strength.5,65. J. Adamcik and R. Mezzenga, “Proteins fibrils from a polymer physics perspective,” Macromolecules 45, 1137–1150 (2012). https://doi.org/10.1021/ma202157h6. Y. Shen, A. Levin, A. Kamada, Z. Toprakcioglu, M. Rodriguez-Garcia, Y. Xu, and T. P. Knowles, “From protein building blocks to functional materials,” ACS Nano 15, 5819–5837 (2021). https://doi.org/10.1021/acsnano.0c08510 Protein nanofibrils are of great interest due to their involvement in neurodegenerative diseases such as Alzheimer’s and Parkinson’s and their applications in food products (e.g., as thickening and gelling agents).6–86. Y. Shen, A. Levin, A. Kamada, Z. Toprakcioglu, M. Rodriguez-Garcia, Y. Xu, and T. P. Knowles, “From protein building blocks to functional materials,” ACS Nano 15, 5819–5837 (2021). https://doi.org/10.1021/acsnano.0c085107. J. Peng, V. Calabrese, S. J. Veen, P. Versluis, K. P. Velikov, P. Venema, and E. van der Linden, “Rheology and microstructure of dispersions of protein fibrils and cellulose microfibrils,” Food Hydrocolloids 82, 196–208 (2018). https://doi.org/10.1016/j.foodhyd.2018.03.0338. S. S. Rogers, P. Venema, L. M. Sagis, E. Van Der Linden, and A. M. Donald, “Measuring the length distribution of a fibril system: A flow birefringence technique applied to amyloid fibrils,” Macromolecules 38, 2948–2958 (2005). https://doi.org/10.1021/ma0474224 Most naturally derived CR that can be produced in mass quantities for industrial purposes have a polydisperse contour length, lc, (the length of the fully stretched CR) and a morphology that can significantly deviate from those of perfect rods [see CNC in Fig. 1(a)]. This implies that no available theory guarantees a reasonable prediction of structural and rheological properties of naturally derived CR dispersions under flow. Since these rod-like particles are suitable building blocks for the development of sustainable materials, establishing an experimental framework to study the dynamics of such CR under flow becomes essential. In contrast, filamentous viruses are a group of naturally produced CR with monodisperse lc and diameter, d, that can be adequately approximated by a cylindrical shape [see Pf1 in Fig. 1(c)]. This makes filamentous viruses particularly suited for testing theories and physics concepts regarding the flow behavior of rod-like particles.11–1611. C. Lang, J. Hendricks, Z. Zhang, N. K. Reddy, J. P. Rothstein, M. P. Lettinga, J. Vermant, and C. Clasen, “Effects of particle stiffness on the extensional rheology of model rod-like nanoparticle suspensions,” Soft Matter 15, 833–841 (2019). https://doi.org/10.1039/C8SM01925H12. C. Lang, J. Kohlbrecher, L. Porcar, A. Radulescu, K. Sellinghoff, J. K. G. Dhont, and M. P. Lettinga, “Microstructural understanding of the length- and stiffness-dependent shear thinning in semidilute colloidal rods,” Macromolecules 52, 9604–9612 (2019). https://doi.org/10.1021/acs.macromol.9b0159213. N. Nemoto, J. L. Schrag, J. D. Ferry, and R. W. Fulton, “Infinite-dilution viscoelastic properties of tobacco mosaic virus,” Biopolymers 14, 409–417 (1975). https://doi.org/10.1002/bip.1975.36014021314. M. A. Kanso, J. H. Piette, J. A. Hanna, and A. J. Giacomin, “Coronavirus rotational diffusivity,” Phys. Fluids 32, 113101 (2020). https://doi.org/10.1063/5.003187515. Erglis, K., V. Ose, A. Zeltins, and A. Cebers, “Viscoelasticity of the bacteriophage Pf1 network measured by magnetic microrheology,” Magnetohydrodynamics 46, 23–29 (2010). https://doi.org/10.22364/mhd.46.1.216. Barry, E., D. Beller, and Z. Dogic, “A model liquid crystalline system based on rodlike viruses with variable chirality and persistence length,” Soft Matter 5, 2563–2570 (2009). https://doi.org/10.1039/b822478a The most used viruses comprise tobacco mosaic virus (TMV), Pseudomonas phage (Pf1), fd [Fig. 1(d)], fdY21M, and M13k07.1717. Z. Dogic and S. Fraden, “Ordered phases of filamentous viruses,” Curr. Opin. Colloid Interface Sci. 11, 47–55 (2006). https://doi.org/10.1016/j.cocis.2005.10.004 In contrast to single polymer chains in solutions, CR are relatively stiff due to the assembly of multiple macromolecules across the diameter. This supramolecular assembly across the CR diameter impedes CR from adopting different conformations arising from the rotational freedom around the backbone. Consequently, CR flexibility arises exclusively from thermal bending fluctuations along the backbone.1818. Z. G. Wang, “50th anniversary perspective: Polymer conformation—A pedagogical review,” Macromolecules 50, 9073–9114 (2017). https://doi.org/10.1021/acs.macromol.7b01518 CR flexibility is typically categorized according to their persistence length lp, a material property related to the elastic modulus (E) and the diameter (d) of the CR, and the thermal energy kbT as lp∝d4E/kbT.1919. I. Usov, G. Nyström, J. Adamcik, S. Handschin, C. Schütz, A. Fall, L. Bergström, and R. Mezzenga, “Understanding nanocellulose chirality and structure–properties relationship at the single fibril level,” Nat. Commun. 6, 7564 (2015). https://doi.org/10.1038/ncomms8564 Practically, the persistence length lp quantifies the length scale below which the CR behave as rigid rod-like segments. CR are commonly considered as rigid (lp≫lc) or semi-flexible (lp∼lc). Compared to high molecular weight polymer chains in solutions, for which lp≪lc, CR are considerably stiffer.The concentration regimes for CR are based on the number of CR per unit volume, referred to as the number density ν=(4ϕv)/(d2lcπ) (in m−3), with ϕv being the volume fraction. CR are isolated from each other in the dilute regime when ν<lc−3 and interacting in the semi-dilute regime, lc−3<ν<d−1lc−2.20,2120. M. Doi, S. F. Edwards, and S. F. Edwards, The Theory of Polymer Dynamics (Oxford University Press, 1988).21. M. J. Solomon and P. T. Spicer, “Microstructural regimes of colloidal rod suspensions, gels, and glasses,” Soft Matter 6, 1391–1400 (2010). https://doi.org/10.1039/b918281k The effective rotational diffusion coefficient Dreff depends on the concentration regime. In the dilute regime, Dreff is given as11,12,2011. C. Lang, J. Hendricks, Z. Zhang, N. K. Reddy, J. P. Rothstein, M. P. Lettinga, J. Vermant, and C. Clasen, “Effects of particle stiffness on the extensional rheology of model rod-like nanoparticle suspensions,” Soft Matter 15, 833–841 (2019). https://doi.org/10.1039/C8SM01925H12. C. Lang, J. Kohlbrecher, L. Porcar, A. Radulescu, K. Sellinghoff, J. K. G. Dhont, and M. P. Lettinga, “Microstructural understanding of the length- and stiffness-dependent shear thinning in semidilute colloidal rods,” Macromolecules 52, 9604–9612 (2019). https://doi.org/10.1021/acs.macromol.9b0159220. M. Doi, S. F. Edwards, and S. F. Edwards, The Theory of Polymer Dynamics (Oxford University Press, 1988). Dreff≡Dr0=3kbTln(lc/d)πηslc3,(1)where kb=1.38×10−23 J/K is the Boltzmann constant, T is the absolute temperature, and ηs is the solvent viscosity. By inspecting Eq. (1), Dr0 is independent of ν, thus Dr0 is predicted to be constant throughout the dilute regime, when interparticle interactions are negligible. In the semi-dilute regime, interparticle interactions lead to an effective rotational diffusion coefficient that decreases with increasing CR concentration (as shown in Fig. 2 for CNC,2222. V. Calabrese, S. Varchanis, S. J. Haward, and A. Q. Shen, “Alignment of colloidal rods in crowded environments,” Macromolecules 55, 5610–5620 (2022). https://doi.org/10.1021/acs.macromol.2c00769 several filamentous viruses23,2423. Barabé, B., S. Abakumov, D. Z. Gunes, and M. P. Lettinga, “Sedimentation of large particles in a suspension of colloidal rods,” Phys. Fluids 32, 053105 (2020). https://doi.org/10.1063/5.000607624. S. Abakumov, O. Deschaume, C. Bartic, C. Lang, O. Korculanin, J. K. G. Dhont, and M. P. Lettinga, “Uncovering log jamming in semidilute suspensions of quasi-ideal rods,” Macromolecules 54, 9609–9617 (2021). https://doi.org/10.1021/acs.macromol.1c00876 and compared with results from numerical simulations).2525. Y. G. Tao, W. K. Den Otter, J. K. Dhont, and W. J. Briels, “Isotropic-nematic spinodals of rigid long thin rodlike colloids by event-driven Brownian dynamics simulations,” J. Chem. Phys. 124, 134906 (2006). https://doi.org/10.1063/1.2180251 According to the Doi–Edwards theory, in the semi-dilute regime,20,2620. M. Doi, S. F. Edwards, and S. F. Edwards, The Theory of Polymer Dynamics (Oxford University Press, 1988).26. M. Doi and S. F. Edwards, “Dynamics of rod-like macromolecules in concentrated solution. Part 1,” J. Chem. Soc. Faraday Trans. 2 74, 560 (1978). https://doi.org/10.1039/f29787400560 Dreff≡Dr=βDr0(νlc3)−2,(2)where β is a dimensionless and length-independent prefactor ≫1 that governs the concentration range where the scaling Dr/Dr0∝(νlc3)−2 is respected. From simulations and theory, it has been shown that β=1.3×103 (see simulation data and theory in Fig. 2).20,25,2720. M. Doi, S. F. Edwards, and S. F. Edwards, The Theory of Polymer Dynamics (Oxford University Press, 1988).25. Y. G. Tao, W. K. Den Otter, J. K. Dhont, and W. J. Briels, “Isotropic-nematic spinodals of rigid long thin rodlike colloids by event-driven Brownian dynamics simulations,” J. Chem. Phys. 124, 134906 (2006). https://doi.org/10.1063/1.218025127. I. Teraoka, N. Ookubo, and R. Hayakawa, “Molecular theory on the entanglement effect of rodlike polymers,” Phys. Rev. Lett. 55, 2712–2715 (1985). https://doi.org/10.1103/PhysRevLett.55.2712 Recently, using filamentous viruses with distinct lc, d, and 0.09<lc/lp<0.75, it has been shown experimentally that 1.1×103<β<2.2×103 (defined by the green area in Fig. 2), comparable to theoretical prediction.1212. C. Lang, J. Kohlbrecher, L. Porcar, A. Radulescu, K. Sellinghoff, J. K. G. Dhont, and M. P. Lettinga, “Microstructural understanding of the length- and stiffness-dependent shear thinning in semidilute colloidal rods,” Macromolecules 52, 9604–9612 (2019). https://doi.org/10.1021/acs.macromol.9b01592 Nonetheless, deviations from the value of β=1.3×103 for ideal rigid rods have been reported.8,288. S. S. Rogers, P. Venema, L. M. Sagis, E. Van Der Linden, and A. M. Donald, “Measuring the length distribution of a fibril system: A flow birefringence technique applied to amyloid fibrils,” Macromolecules 38, 2948–2958 (2005). https://doi.org/10.1021/ma047422428. C. Brouzet, N. Mittal, F. Lundell, and L. D. Söderberg, “Characterizing the orientational and network dynamics of polydisperse nanofibers on the nanoscale,” Macromolecules 52, 2286–2295 (2019). https://doi.org/10.1021/acs.macromol.8b02714 A deviation from the theoretical value can be linked to multiple factors. For instance, the selection of a single representative value for lc in a polydisperse CR population, the assessment of d and the density of the CR (affecting ν), interparticle attractive forces, and CR flexibility.For ν>d−1lc−2, CR are in the concentrated regime and their diffusion becomes severely hindered as excluded volume interactions become important. In the concentrated regime, rigid rod-like particles can adopt a favorable orientation at rest, a phase that is referred to as the liquid crystal phase.20,2120. M. Doi, S. F. Edwards, and S. F. Edwards, The Theory of Polymer Dynamics (Oxford University Press, 1988).21. M. J. Solomon and P. T. Spicer, “Microstructural regimes of colloidal rod suspensions, gels, and glasses,” Soft Matter 6, 1391–1400 (2010). https://doi.org/10.1039/b918281kThe correct estimation of the effective rotational diffusion coefficient Dreff is important because it sets the value of the characteristic deformation rate E (i.e., E≡γ˙ for the shear rate and E≡ε˙ for the extensional rate, respectively) required to induce the alignment of the CR. Specifically, the Péclet number Pe=|E|/Dreff indicates the relative importance of the hydrodynamic forces over rotational Brownian diffusion. For Pe<1, colloidal rods are isotropically distributed via Brownian diffusion while for Pe≳1, hydrodynamic forces are sufficiently strong to favor alignment of the rods. Additionally, Dreff gives the relaxation time of the CR as τ=1/(6Dreff).2020. M. Doi, S. F. Edwards, and S. F. Edwards, The Theory of Polymer Dynamics (Oxford University Press, 1988). The estimation of τ is pivotal for the synthesis of structurally ordered materials that rely on the flow-induced alignment of CR. For instance, to retain out-of-equilibrium CR orientations, physical or chemical processes able to lock the microstructure in place must occur within a time scale of ≪τ.

Microfluidic devices are versatile platforms to study the flow-induced alignment of CR in different flow fields and for the retrieval of relevant time- and length-scales associated with the CR. For instance, microfluidic devices can be designed to generate two-dimensional (2D) flows that provide, to a good approximation, a uniform flow through the channel height (e.g., microfluidics with rectangular cross sections and relatively high aspect ratio). 2D flows are convenient for flow visualization and for techniques aimed at probing the structural properties of the fluid, such as flow-induced birefringence (FIB) and small-angle scattering (SAS). Alternatively, three-dimensional (3D) flows, generated in microfluidic devices with symmetric and/or low aspect ratio cross-sections, are more representative of processing condition under which naturally derived CR may be employed (e.g., fiber spinning). Most importantly, microfluidics can be designed to generate not only shearing flows (the flow type generated by rotational rheometers) but also extensional flows and mixed flows comprising both shearing and extensional deformations. Thus, coupling microfluidics with flow visualization techniques, pressure sensors, FIB, and/or SAS techniques provides a comprehensive fingerprint of the structure–property relationship of colloidal rods in unique flow scenarios. Moreover, the small length scales adopted in microfluidic devices allow relatively high deformation rates while preserving creeping flow conditions.

This article provides an overview of microfluidic-based techniques to study the dynamics of naturally derived CR. Particular attention is given to CR in concentrations below the liquid crystal phase, where CR are isotropically distributed at rest but can adopt a favorable orientation under flow. In the following sections, we discuss some of the latest results regarding CR and relevant rod-like macromolecules under shearing and extensional flows.

II. SHEARING FLOW

Section:

ChooseTop of pageABSTRACTI. INTRODUCTIONII. SHEARING FLOW <<III. EXTENSIONAL FLOWIV. CONCLUSIONS AND OUTLO...REFERENCESUnderstanding the flow-induced alignment of rod-like particles is pivotal for developing materials with structural anisotropy. Although mixed flows are present in real-life applications, studying the shear-induced alignment of CR can provide insightful information that can be interpreted and analyzed based on existing literature and be directly correlated with established rheological parameters (e.g., steady shear viscosity and steady shear stress). In our research group, we have used glass microfluidic devices, fabricated by selective laser-induced etching, with high aspect ratio [H/W=5, where H=2mm and W=0.4mm are the channel height and width, respectively, Figs. 3(a) and 3(b)] to generate good approximations of 2D purely shearing flows.30,3130. N. Burshtein, S. T. Chan, K. Toda-Peters, A. Q. Shen, and S. J. Haward, “3D-printed glass microfluidics for fluid dynamics and rheology,” Curr. Opin. Colloid Interface Sci. 43, 1–14 (2019). https://doi.org/10.1016/j.cocis.2018.12.00531. S. J. Haward, K. Toda-Peters, and A. Q. Shen, “Steady viscoelastic flow around high-aspect-ratio, low-blockage-ratio microfluidic cylinders,” J. Non-Newtonian Fluid Mech. 254, 23–35 (2018). https://doi.org/10.1016/j.jnnfm.2018.02.009 Using a straight microfluidic device with a rectangular cross section [Figs. 3(a) and 3(b)], we have probed the alignment of CR such as CNC, worm-like block copolymers, and protein nanofibrils in the flow-velocity gradient plane using FIB.9,22,32–349. V. Calabrese, S. J. Haward, and A. Q. Shen, “Effects of shearing and extensional flows on the alignment of colloidal rods,” Macromolecules 54, 4176–4185 (2021). https://doi.org/10.1021/acs.macromol.0c0215522. V. Calabrese, S. Varchanis, S. J. Haward, and A. Q. Shen, “Alignment of colloidal rods in crowded environments,” Macromolecules 55, 5610–5620 (2022). https://doi.org/10.1021/acs.macromol.2c0076932. V. Calabrese, S. Varchanis, S. J. Haward, J. Tsamopoulos, and A. Q. Shen, “Structure-property relationship of a soft colloidal glass in simple and mixed flows,” J. Colloid Interface Sci. 601, 454–466 (2021). https://doi.org/10.1016/j.jcis.2021.05.10333. V. Calabrese, C. György, S. J. Haward, T. J. Neal, S. P. Armes, and A. Q. Shen, “Microstructural dynamics and rheology of worm-like diblock copolymer nanoparticle dispersions under a simple shear and a planar extensional flow,” Macromolecules 55, 10031–10042 (2022). https://doi.org/10.1021/acs.macromol.2c0131434. T. P. Santos, V. Calabrese, M. W. Boehm, S. K. Baier, and A. Q. Shen, “Flow-induced alignment of protein nanofibril dispersions,” J. Colloid Interface Sci. 638, 487–497 (2023). https://doi.org/10.1016/j.jcis.2023.01.105 FIB provides two parameters related to the structural anisotropy of the fluid, namely, the birefringence, Δn, and the orientation of the slow optical axis, θ. For randomly oriented CR dispersed in an isotropic medium (e.g., water), Δn=0, while flow-induced alignment of CR is marked by Δn>0. CR are usually optically positive, so that the orientation of the slow optical axis, θ, captures the direction of orientation of the rods.3535. C. Lane, D. Rode, and T. Rösgen, “Birefringent properties of aqueous cellulose nanocrystal suspensions,” Cellulose 29, 6093–6107 (2022). https://doi.org/10.1007/s10570-022-04646-y For each volumetric flow rate (Q, m3/s), FIB provides the spatial distribution of the birefringence, Δn, [shown in the normalized form as Δn/ϕ, where ϕ is the mass fraction of the CR, in Fig. 3(c)] and the orientation of the slow optical axis, θ [Fig. 3(d)]. Analogous mapping of the flow-induced orientation of CR can be retrieved using scanning-small-angle x-ray scattering, referred to as scanning-SAXS.36–3936. V. Lutz-Bueno, J. Zhao, R. Mezzenga, T. Pfohl, P. Fischer, and M. Liebi, “Scanning-SAXS of microfluidic flows: Nanostructural mapping of soft matter,” Lab Chip 16, 4028–4035 (2016). https://doi.org/10.1039/C6LC00690F37. Corona, P. T., B. Berke, M. Guizar-Sicairos, L. G. Leal, M. Liebi, and M. E. Helgeson, “Fingerprinting soft material nanostructure response to complex flow histories,” Phys. Rev. Mater. 6, 1–14 (2022). https://doi.org/10.1103/PhysRevMaterials.6.04560338. S. J. S. Qazi, A. R. Rennie, I. Tucker, J. Penfold, and I. Grillo, “Alignment of dispersions of plate-like colloidal particles of Ni(OH)2 induced by elongational flow,” J. Phys. Chem. B 115, 3271–3280 (2011). https://doi.org/10.1021/jp108805m39. Rosén, T., R. Wang, H. R. He, C. Zhan, S. Chodankar, and B. S. Hsiao, “Shear-free mixing to achieve accurate temporospatial nanoscale kinetics through scanning-SAXS: Ion-induced phase transition of dispersed cellulose nanocrystals,” Lab Chip 21, 1084–1095 (2021). https://doi.org/10.1039/D0LC01048K For scanning-SAXS, each scattering pattern is acquired along the region of interest point by point, making typical acquisition times long compared to FIB. For instance, for a dispersion of protein nanofibers, scanning-SAXS required an acquisition time of ∼5 min to scan 1 mm2 for a single flow rate, while at most a few seconds would be required for a typical FIB experiment.3636. V. Lutz-Bueno, J. Zhao, R. Mezzenga, T. Pfohl, P. Fischer, and M. Liebi, “Scanning-SAXS of microfluidic flows: Nanostructural mapping of soft matter,” Lab Chip 16, 4028–4035 (2016). https://doi.org/10.1039/C6LC00690F Importantly, the birefringence, Δn, and the order parameter, S, obtained from small-angle scattering techniques (computed from the anisotropy of the 2D scattering patterns) provide comparable information of the CR orientation under flow as Δn∝S.40–4240. K. R. Purdy, Z. Dogic, S. Fraden, A. Rühm, L. Lurio, and S. G. J. Mochrie, “Measuring the nematic order of suspensions of colloidal fd virus by x-ray diffraction and optical birefringence,” Phys. Rev. E 67, 031708 (2003). https://doi.org/10.1103/PhysRevE.67.03170841. K. Uetani, H. Koga, and M. Nogi, “Estimation of the intrinsic birefringence of cellulose using bacterial cellulose nanofiber films,” ACS Macro Lett. 8, 250–254 (2019). https://doi.org/10.1021/acsmacrolett.9b0002442. Rosén, T., R. Wang, C. Zhan, H. He, S. Chodankar, and B. S. Hsiao, “Cellulose nanofibrils and nanocrystals in confined flow: Single-particle dynamics to collective alignment revealed through scanning small-angle x-ray scattering and numerical simulations,” Phys. Rev. E 101, 032610 (2020). https://doi.org/10.1103/PhysRevE.101.032610 On the other hand, compared to FIB, SAS can provide additional information regarding the CR morphology and particle–particle interactions within the same experiment when performed on the sample at rest. For quantitative comparisons of Δn and θ, it is convenient to spatially average these quantities (denoted as ⟨Δn⟩ and ⟨θ⟩) across a location of the microfluidic device where the deformation rate is constant and to plot them as a function of the respective deformation rate. For instance, in our previous publications using the microfluidic design given in Figs. 3(a) and 3(b), at each flow rate, we averaged Δn along a section of the microchannel located at y=±0.1 mm [see dashed lines sketched in Figs. 3(c) and 3(d)].9,22,339. V. Calabrese, S. J. Haward, and A. Q. Shen, “Effects of shearing and extensional flows on the alignment of colloidal rods,” Macromolecules 54, 4176–4185 (2021). https://doi.org/10.1021/acs.macromol.0c0215522. V. Calabrese, S. Varchanis, S. J. Haward, and A. Q. Shen, “Alignment of colloidal rods in crowded environments,” Macromolecules 55, 5610–5620 (2022). https://doi.org/10.1021/acs.macromol.2c0076933. V. Calabrese, C. György, S. J. Haward, T. J. Neal, S. P. Armes, and A. Q. Shen, “Microstructural dynamics and rheology of worm-like diblock copolymer nanoparticle dispersions under a simple shear and a planar extensional flow,” Macromolecules 55, 10031–10042 (2022). https://doi.org/10.1021/acs.macromol.2c01314 These specific locations were chosen because they correspond to the middle distance between the centerline (y=0 mm) and the channel wall, thus in between the location of the lowest and highest shear rate, respectively. Once the deformation rate at specific channel location is obtained [e.g., experimentally using micro-particle image velocimetry (μ-PIV) or obtained computationally by choosing a suitable rheological model], the spatially averaged birefringence, ⟨Δn⟩, and the spatially averaged slow orientation angle, ⟨θ⟩, vs |γ˙| can be constructed, as shown for a semi-dilute CNC dispersion [left-side graphs in Fig. 3(e)]. Since the birefringence, Δn, is directly linked to the number of aligned CR segments along the optical path, the critical shear rate at the onset of birefringence (|γ˙∗|) provides a good estimation of Dreff since |γ˙∗|=Dreff (i.e., Pe=1). For a shear flow, the orientation angle of the CR with respect to the flow direction in the flow velocity-gradient plane can be generally described as ⟨θ⟩=π4−12arctan[(|γ˙|6Dreff)α],(3)where 0<α≤1 is a stretching exponent that accounts for particle polydispersity.43–4643. N. K. Reddy, G. Natale, R. K. Prud’homme, and J. Vermant, “Rheo-optical analysis of functionalized graphene suspensions,” Langmuir 34, 7844–7851 (2018). https://doi.org/10.1021/acs.langmuir.8b0157444. C. P. Lindsey and G. D. Patterson, “Detailed comparison of the Williams-Watts and Cole-Davidson functions,” J. Chem. Phys. 73, 3348–3357 (1980). https://doi.org/10.1063/1.440530