Extensibility governs the flow-induced alignment of polymers and rod-like colloids

Polymers and rod-like colloids (PaRC) adopt a favorable orientation under sufficiently strong flows. However, how the flow kinematics affect the alignment of such nanostructures according to their extensibility remains unclear. By analysing the shear- and extension-induced alignment of chemically and structurally different PaRC, we show that extensibility is a key determinant of the structural response to the imposed kinematics. We propose a unified description of the effectiveness of extensional flow, compared to shearing flow, at aligning PaRC of different extensibility.

Polymers and rod-like colloids (PaRC) are ubiquitous in biological fluids (e.g., mucus, saliva), food and industrial formulations (e.g., gels, paints), imparting specific properties and functionalities.When solubilized or dispersed in a solvent, (PaRC) adopt an equilibrium conformation (e.g., rod-like, worm-like, coil-like) that depends on a multitude of factors including surface chemistry, contour length, and backbone rigidity, affecting the PaRC extensibility and flexibility [1].The PaRC extensibility (L e ) describes the ratio between the contour length of the fully extended structure (l c ) and the respective size at equilibrium.The PaRC flexibility (N p ) represents the number of persistence length (l p ) segments composing the PaRC contour length [1,2].For rigid colloidal rods L e → 1 and N p ≪ 1, whilst for flexible polymers that adopt a coil-like conformation at equilibrium, L e ≫ 1 and N p ≫ 1.Under sufficiently strong imposed flows, PaRC are driven out of equilibrium and towards a state of alignment induced by velocity gradients in the flow field.Flow-induced alignment of PaRC is central to processes including fiber spinning [3], bacteriophage replication [4,5], and amyloid fibrillogenesis related to neurodegenerative diseases [6,7].The link between the PaRC conformation and the bulk fluid properties at the equilibrium, such as the zero-shear viscosity and the macromolecular time scales of diffusion and relaxation, are well understood and described by generalized scaling theories [1,8,2].Nonetheless, how the dynamics of PaRC under flow depend on their equilibrium conformation remains elusive.It is known that rigid colloidal rods tend to adopt a favourable orientation under characteristic deformation rates |E| able to overcome the rotational Brownian diffusion [2,9,10].This condition is captured by the Péclet number P e = 6|E|τ ≳ 1, with τ the longest relaxation time of the rigid rod [2,11].For sufficiently strong shear flows (P e ≳ 1), rigid colloidal rods orient but also undergo occasional tumbling due to the vorticity component of the strain rate tensor [12,13].On the contrary, pure extensional flows are vorticity-free and generally regarded as more effective than shear flows at inducing the alignment of rigid colloidal rods [14,15,16,17,9].For flexible polymers in solution, favourable chain orientation is expected to occur at Weissenberg number W i = |E|τ ≳ 0.5, with τ the longest relaxation time of the polymer [18,19,20,21].Extensional flows are considered pivotal at inducing a preferential orientation of flexible polymers.Direct single molecule imaging of λ-DNA (L e ∼ 13) has shown that extensional flows provide a greater extent of chain extension than shear flows [22,23].For solutions of even more highly flexible synthetic polymers (e.g., poly(ethylene oxide) and atactic-polystyrene with L e > 50) microfluidic studies reporting flow-induced birefringence (FIB) show that macromolecular alignment is dominant in localized regions originating from hyperbolic points in the flow field where purely extensional kinematics are approximated [24,25,19,26].
In this letter we investigate the steady shear-and extension-induced alignment of PaRC with varying extensibility, including rod-like colloids and significantly more flexible polyelectrolytes in solution.Employing quantitative FIB imaging in a specific microfluidic device, we show that different deformation rates are required for the onset of PaRC alignment in shear and extension, and that the difference is correlated with the PaRC extensibility.In the limit of rigid rod-like nanostructures we validate our results with the revised Doi-Edwards theory proposed by Lang et al. [10] Our study aims towards unifying the relationship between shear-and extension-induced alignment of PaRC with various extensibility.
PaRC alignment is evaluated via FIB imaging performed on a CRYSTA PI-1P camera (Photron Ltd, Japan) fitted with a 4× Nikon objective lens.Microparticle image velocimetry (µPIV) (TSI Inc., MN, 4× and 10× Nikon Objectives) is used to characterize the nature of the imposed flow kinematics in the microfluidic device [27].The microfluidic device (Fig. 1(a)), fabricated in fused silica glass [28], is a numerically-optimized version of the cross-slot device, referred to as the optimized shape cross-slot extensional rheometer (OSCER) [24,25].The device has an aspect ratio H/W = 10, where H = 3 mm is the height, defining the optical path along the z−axis, and W = 0.3 mm is the width (in the x-y plane), producing a good approximation to a two dimensional flow.FIB and µPIV are evaluated at the x-y plane in two distinct regions of interest (ROI) of the microfluidic device generating extension-or shear-dominated flows (marked by the coloured boxes in Fig. 1(a)).In the extension-dominated ROI (magenta box, Fig. 1(a)), FIB and µPIV are evaluated around the stagnation point (i.e., x = y = 0 mm) where a constant and shear-free planar extension is generated when opposing inlets (I 1 , I 2 ) and outlets (O 1 , O 2 ) operate at equal and opposite volumetric flow rates (Q m 3 /s) (representative flow field in Fig. 1 CMC with Mw of 0.4, 0.6 and 0.8 MDa, are referred to as CMC04, CMC06 and CMC08, respectively, and HA with Mw of 0.9, 1.6, 2.6 and 4.8 MDa, are referred to as HA09, HA16, HA26 and HA48, respectively [29].For the rod-like colloids, the characteristic contour length ⟨l c ⟩ and the persistence length l p are obtained from atomic force microscopy (AFM) [30,27].For CMC and HA, l p is obtained from literature [31,32,33], and ⟨l c ⟩ = N a, where N is the is the weight averaged degree of polymerisation, and a is the monomer length.
which approaches the scaling L e ∝ N 0.5 p for large N p .FIB patterns in the extension-and shear-dominated ROI acquired at an equal average flow velocity |U | are shown in Fig. 2 for representative PaRC with distinct L e .For each system and ROI investigated, the birefringence (∆n), probing the extent of fluid anisotropy due to the PaRC alignment, is normalized by the maximum birefringence (∆n max ).For fd and Pf1 a diamond-like pattern of intense birefringence is shown in the extension-dominated ROI as previously shown for colloidal rods with L e ∼ 1 [14,15].For the comparatively more extensible CMC08 (L e ∼ 7), the diamond-like birefringence pattern fades and an intense birefringence strand develops along the extensional axis (i.e., x).For the most extensible PaRC investigated, HA48 (L e ∼ 16), the diamond-like birefringence pattern disappears and only the intense birefringence strand becomes visible.Fluid elements accumulate strain exponentially as they enter the extensional ROI and approach the extension axis [25].Thus, more extensible PaRC need to pass close to the stagnation point to accumulate enough strain to stretch, leading to a localized birefringence strand.Contrarily, more rigid rod-like PaRC require a relatively small accumulated strain to align, leading to a significant alignment even away from the stagnation point, resulting in a diamond-like birefringence pattern.In the shear-dominated ROI, the birefringence increases from the centerline to the channel walls, in qualitative agreement with the trend of shear rate | γ| retrieved from µPIV [27].To quantify the difference between shear-and extension-induced alignment of PaRC with distinct extensibility, we spatially average the birefringence (⟨∆n⟩) in specific locations of the shear-and extension-dominated ROI where the shear and extension deformation rates are constant.In the extension-dominated ROI, the birefringence is averaged along 1 mm of the extension axis (i.e., at y = 0 for −0.5 < x < 0.5, dotted line in Fig. 2(a), top panel) and plotted as a function of the extension rate | ε| along the same line (determined from the flow field measured by µPIV).In the shear-dominated ROI, the birefringence is averaged along the x direction at y = ±W/4 (marked by the dotted line in Fig. 2(a), bottom panel) and plotted as a function of the respective shear rate at the same location (| γ|) [27].We note that the same flow strength for shear and planar extension is given by the magnitude of the strain rate tensor as | γ| = 2| ε|.
In Fig. 3 we plot the normalized birefringence, ⟨∆n⟩/ϕ, with ϕ the mass fraction of the PaRC, as a function of | γ| (filled symbols) and | ε| (empty symbols) for three representative PaRC with distinct L e .For the relatively rigid fd (L e ∼ 1), the onset of birefringence occurs at lower values of | ε| than | γ|, indicating that fd alignment is induced more readily by extension than by shear (Fig. 3(a)).We introduce a non-dimensional scaling factor (SF ) to the extension rate as SF | ε| to match the onset of birefringence in extension with that in shear.Practically, SF provides an estimate of the ratio between the critical shear rate (| γ * |) and critical extension rate For fd, SF = 2.3 captures the difference between extension-and shear-induced alignment at low deformation rates (Fig. 3(b)).The scaling procedure highlights the difference at high deformation rates where the birefringence in extension reaches a greater plateau value than observed in shear.Using the known dimensions of the fd virus, we compare our experimental results with the revised Doi-Edwards theory for ideal, rigid and monodisperse rods proposed by Lang et al. [10,27] The theoretical projected order parameter ⟨P 2 ⟩ is compared with the experimentally measured birefringence as ⟨P 2 ⟩∆n 0 = ⟨∆n⟩/ϕ, where ∆n 0 is the birefringence of perfectly aligned PaRC at ϕ = 1 [35,36,37].To provide a close comparison between theory and the experimental data, in Fig. 3(a, b) the theoretical curves are scaled by ∆n 0 = 0.015, a value comparable to a previously reported value for fd [35].The theory predicts well the birefringence increase with a slope of 1 in both shear and in extension and also predicts the saturation of birefringence at high deformation rates (Fig. 3(b)).The theory confirms that (i) a lower extension rate is required to induce fd alignment compared to the shear rate, and that (ii) at high deformation rates the greatest extent of alignment occurs in extensional flow.Both (i) and (ii) are consistent with the absence of tumbling events in extension, enhancing the overall extent of alignment.For the theoretical curves, SF = 2 (i.e., comparing equal flow strength) enables superimposition of the birefringence curves at low deformation rates in a similar fashion as for the fd dispersion (Fig. 3(b)).Thus, on the base of theory SF = 2 is expected in the limiting case of rigid rod-like nanostructures (L e = 1).The slight difference between SF from the experiment (e.g., SF = 2.3 for fd) and theory is potentially associated with the fact that fd is not perfectly rigid as assumed by the theory (see inset AFM image in Fig. 3(a)).
With progressively increasing extensibility, the difference between | ε * | and | γ * | becomes more pronounced, as shown for Pf1 (L e ≈ 2, Fig. 3(c, d)) and ct-DNA (L e ≈ 10, Fig. 3(e, f)).In these cases, scaling factors of SF = 4 and SF = 12 are required, respectively, to match the onset of birefringence.These results suggest that with the increasing PaRC extensibility, extensional deformations become progressively more effective at inducing the onset of PaRC alignment than shear deformations.Since the effectiveness of the extensional deformations relative to shear deformations at inducing PaRC alignment is captured by SF , we compare the SF for a library of PaRC with distinct L e (Fig. 4(a)).Additionally, in Fig. 4(b) and (c) we plot SF as a function of the reduced extensibility parameter Le = (⟨l c ⟩ − ⟨R 2 ⟩)/ ⟨R 2 ⟩ = L e − 1, to highlight the differences for stiffer PaRC, and as a function of N p , respectively.Notwithstanding the different chemical structures and architectures of the PaRC investigated, a general trend of SF as a function of L e emerges.Specifically, the SF increase with L e can be approximated as SF = 2L β e , where β = 2/3 is a dimensionless parameter (blue line, Fig. 4(a)).This function also captures the main features for the SF as a function of Le and N p in Fig. 4

Figure 1 :
Figure 1: (a) A photograph of the OSCER device with the extension-and shear-dominated ROI marked by the magenta and red box, respectively.Flow fields for a Newtonian fluid at Re < 1 in the extension-and shear-dominated ROI in (b) and (c), respectively.The measured velocity magnitude |V | is scaled by the average flow velocity |U |.
[27]  For ct-DNA, ⟨l c ⟩ is obtained by multiplying the number of base pairs with their average separation distance (0.34 nm) and l p = 50 nm[20,34].The concentrations of the PaRC are chosen to be as dilute as possible yet sufficient to provide a detectable birefringence signal.Where required, the relaxation time of PaRC, τ , is increased by adding glycerol or sucrose to the acqueous solvent, shifting the onset of PaRC alignment within the experimental window of accessible deformation rates.The extensibility of the PaRC is computed as L e = ⟨l c ⟩/ ⟨R 2 ⟩ where ⟨R 2 ⟩ = 2l p ⟨l c ⟩ − 2l 2 p (1 − exp(−⟨l c ⟩/l p )) is the mean-squared end-to-end length of PaRC at equilibrium using a worm-like chain model [1].The critical length scale ℓ below which PaRC follow a rod-like behaviour (i.e., ⟨R 2 ⟩ ∝ ℓ) denotes the persistence length l p .The PaRC flexibility N p = ⟨l c ⟩/l p , is related to L e via:
(b) and (c), respectively.The trend of SF as a function of N p shows a plateau for N p < 1 and a transition towards SF ∝ N 1/3 p at N p ≳ 10 (Fig. 4(b)).We physically interpret the evolution of SF as a function of L e , Le and N p based on the flow-induced conformational changes occurring to the PaRC under the

Figure 4 : 2 N 2 p 2 (
Figure 4: SF as a function of L e , Le and N p , in (a), (b) and (c), respectively.The solid line in (a) is describes SF = 2L β e .In (b) the solid line is the plot of SF = 2( Le + 1) β and in (c) is the plot of SF = 2