Alignment of Colloidal Rods in Crowded Environments.

Understanding the hydrodynamic alignment of colloidal rods in polymer solutions is pivotal for manufacturing structurally ordered materials. How polymer crowding influences the flow-induced alignment of suspended colloidal rods remains unclear when rods and polymers share similar length scales. We tackle this problem by analyzing the alignment of colloidal rods suspended in crowded polymer solutions and comparing that to the case where crowding is provided by additional colloidal rods in a pure solvent. We find that the polymer dynamics govern the onset of shear-induced alignment of colloidal rods suspended in polymer solutions, and the control parameter for the alignment of rods is the Weissenberg number, quantifying the elastic response of the polymer to an imposed flow. Moreover, we show that the increasing colloidal alignment with the shear rate follows a universal trend that is independent of the surrounding crowding environment. Our results indicate that colloidal rod alignment in polymer solutions can be predicted on the basis of the critical shear rate at which polymer coils are deformed by the flow, aiding the synthesis and design of anisotropic materials.

Introduction

The ability to control the hydrodynamic alignment of colloidal rods is critical to produce structurally ordered soft materials that possess desirable mechanical, thermal, optical, and electrical properties.[1−8] These anisotropic materials are promising in applications ranging from electronic sensors and soft robotics to tissue engineering and biomedical devices.[5,9−11] In materials science and engineering, colloidal rods are used in combination with other polymers that impart specific functionality to the final composite material (e.g., increasing ductility and mitigating embrittlement).[1,12,13] As such, understanding and controlling the hydrodynamic alignment of colloidal rods in polymer matrixes becomes of pivotal importance in large-scale processing operations.

The existing literature has shown that the most important control parameter for the onset of hydrodynamic alignment of rigid colloidal rods is the Péclet number , a dimensionless number quantifying the relative strength between the imposed deformation rate (e.g., the shear rate (γ̇) and the rotational diffusion coefficient of the rods (Dr)).[14−18] In dilute suspensions, the Péclet number can be defined as , with the rotational diffusion coefficient of the rods given aswhere d and l are the hydrodynamic diameter and length of the colloidal rod, respectively, kb = 1.38 × 10–23 J/K is the Boltzmann constant, T is the absolute temperature, and ηs is the solvent viscosity. For Pe0 < 1, Brownian flocculation dominates, whereas at Pe0 ≥ 1, convective forces are strong enough to induce alignment of the colloidal rods in the flow direction. However, this criterion is valid only under the assumption that the colloidal rods perceive the surrounding fluid as a continuum medium, that is, the characteristic length scale of the colloidal rods, such as the radius of gyration , must be much larger than that associated with the suspending medium.[19−21] Colloidal rods suspended in low Mw solvents such as water generally satisfy this assumption. However, in many industrial and biological processes, colloidal rods flow in crowded environments of polymers in solution where the characteristic length scale of suspended rods is similar to those of the surrounding macromolecules, for example, the polymer radius of gyration, , or the polymer mesh size, ξp (also referred to as the correlation length).[20,22−27] In this scenario, the continuum assumption breaks down, and the rods experience a local viscosity (ηlocal) that lies between the solvent viscosity and the bulk viscosity of the polymeric solution.[22−24,27] In principle, with the knowledge of the value of ηlocal, it is possible to predict the shear rate for the onset of colloidal alignment based on the criterion of Pe0 = 1 using ηlocal in place of ηs in eq . However, the main hurdle in predicting the alignment of colloidal rods suspended in macromolecular solutions based on the criterion Pe0 = 1 stems from the fact that the value of ηlocal is not known a priori. Consequently, to date, it is challenging to predict the onset of flow-induced alignment of colloidal rods with length scales comparable to those of the suspending polymeric fluids.

The flow-induced alignment of rigid elongated particles suspended in viscoelastic polymeric solutions has been studied experimentally and numerically for particles with relevant length scales larger than that associated with the suspending fluid, thus, the suspending polymeric fluid was considered as a continuum medium for the colloidal rods.[26,28−32] In this length-scale context, theories predict a critical deformation rate above which the elastic forces of the suspending fluid cause the particle alignment to drift from the flow direction to the vorticity direction.[26,28,33,34] However, contrasting experimental results have been reported for relatively small particles suspended in shear-thinning polymeric fluids.[26,29,30] For instance, elongated hematite particles (with l = 600 nm) in entangled poly(ethylene oxide) solutions[30] displayed the particle alignment in the vorticity direction, as expected by theory. However, shorter hematite particles (with l = 360 nm) suspended in entangled polystyrene solutions[26] did not, casting doubts on the validity of continuum theories in conditions where colloids and polymers have similar characteristic length scales.

In this work, we elucidate the mechanism driving the onset of colloidal rod alignment in semidilute polymer solutions where polymers act as the crowding agents to tracer colloidal rods. We use cellulose nanocrystals (CNC) as rigid colloidal rods as they are widely used in the synthesis of composite polymer materials with high performance and functionalities.[13,35,36] To understand the effect of crowding on the CNC alignment, we adopt two approaches: (i) increasing the CNC mass fraction, ϕ, in an aqueous Newtonian solvent spanning from the dilute regime where interparticle interactions are negligible up to the semidilute regime where interparticle interactions are at play, providing “self-crowding” of the CNC by other analogous particles; (ii) using shear-thinning (non-Newtonian) polymer solutions as the suspending fluid while keeping ϕ = 1.0 × 10–3 so that the CNC is in the dilute regime with negligible interparticle interactions, and the confinement acting on the CNC is provided only by the surrounding polymer chains, which we refer to as “polymer crowding”. Specifically, we use high Mw neutral polymers (poly(ethylene oxide), PEO, and polyacrylamide, PA), with a radius of gyration Rgp comparable to the CNC length scale (i.e., .[23,37−40] For the polymer crowding case, we show that the onset of CNC alignment is linked with the relaxation time of the polymer solution (τp). Specifically, we show that the Weissenberg number , quantifying the strength of the elastic response of the fluid to an imposed deformation rate, controls the onset of CNC alignment in polymeric media.

Experimental Section

Test Fluids

The test fluids were prepared using an aqueous CNC stock suspension (CelluForce, Montreal, Canada, pH 6.3 at 5.6 wt %). The CNC has an average length ⟨l⟩ = 260 ± 180 nm, a maximum length lmax = 700 nm, an average diameter ⟨d⟩ = 4.8 ± 1.8 nm as detected from atomic force microscopy.[18] The CNC suspensions are electrostatically stabilized when suspended in deionized water with zero salt as a result of the presence of sulfate ester groups.[36,41] Therefore, the CNC bears a negative charge for a wide pH range, with a zeta potential of approximately −64 mV when suspended in deionized water at pH ≈7.[41] The effective hydrodynamic diameter is computed as d = δ + ⟨d⟩ = 27.4 nm, considering the estimated contribution of the electric double layer δ = 22.6 nm in deionized water.[41] Assuming a cylindrical shape, the number density of the CNC was calculated as ν = (4ϕvolume)/(⟨d⟩2lπ), where l is the hydrodynamic length of the CNC obtained experimentally through (specified in the main text) and ϕvolume is the volume fraction of the suspended CNC (calculated using a CNC density of 1560 kg/m3).[42] CNC suspensions at different mass fractions, ϕ, were prepared by dilution of the mother CNC stock with deionized water and mixed on a laboratory roller for at least 24 h at ∼22 °C. Where specified, the ϕ = 1 × 10–3 CNC suspension was prepared in a Newtonian solvent composed of a glycerol/water mixture containing 17.2 vol % glycerol (Sigma-Aldrich 99% with ηs = 1.7 mPa s as measured via shear rheometry). From previous small-angle X-ray scattering studies of CNC suspensions from the same source as that used in the present work, we expect that, for ϕ < 5 × 10–3, interparticle interactions are minimal.[43] From a geometrical argument, the rods are in the dilute regime for ν < 1/l3. Considering ⟨l⟩ = 260 ± 180 nm the representative length, we expect the transition from dilute to semidilute to occur at ϕ ∼ 2 × 10–3.

PEO with Mw ≈ 4 MDa, PEO with Mw ≈ 8 MDa, and PA with Mw ≈ 5.5 MDa, referred to as PEO4, PEO8, and PA5, respectively, were purchased from Sigma-Aldrich in powder form and solubilized in deionized water on a laboratory roller for at least 48 h at ∼22 °C (stock solution). Polymer solutions at different concentrations (c in mg/mL) containing a constant amount of CNC were prepared by diluting the polymer stock solution with deionized water followed by diluting the CNC stock suspension to a final ϕ = 1 × 10–3 and then mixing the solution on a laboratory roller for 24 h at ∼22 °C. Polymer solutions without the CNC were prepared by following the same procedure described above. The polymer concentration where polymers begin to overlap was estimated as , where NA is Avogadro’s number.[38,44] The radius of gyration for the PEO4 and PEO8 was 135 and 202 nm, respectively, estimated as = 0.02Mw0.58.[37,38] For PA5, = (7.5 × 10–3)Mw0.64 = 154 nm.[39] For the PEO8, the polymer concentrations tested spanned between the dilute (c/c* < 1) and semidilute unentangled regime, where c/c* > 1 and the tube diameter , where ϕp is the polymer volume fraction and bk ≈ 3.7 nm is the experimentally determined tube diameter in a PEO melt.[23,40,45] For PEO8 with = 202 nm, the highest polymer concentration tested is c/c* = 10.5, which yields dt ≈ 250 nm, thus in the semidilute unentangled regime (c/c* > 1 and dt > ). The ratio between the average CNC length and the PEO persistence length (lp ≈ 0.5 nm)[46] is ⟨l⟩/lp ∼ 500. The PEO is considered a compatible, nonadsorbing polymer for the CNC.[13,47] In the range of CNC and polymer concentrations tested, the CNC did not show any sign of aggregation, even after a few months from the sample preparation.

Methods

To assess the CNC alignment in crowded environments, we measured the flow-induced birefringence (FIB) as a function of the shear rate in a straight microfluidic channel etched in fused silica (Figure a,b). The microfluidic channel has a rectangular cross section with height H = 2 mm along the z-axis, corresponding to the optical path, and width W = 0.4 mm, along the y-axis, thus providing an approximation to two-dimension (2D).[18,48] The flow in the microfluidic channel is driven along the channel length (x-axis) by a syringe pump (Cetoni Nemesys) and Hamilton Gastight syringes infusing and withdrawing at an equal and opposite volumetric flow rate (Q) from the inlet and outlet, respectively. The average flow velocity in the channel is U = Q(WH)−1.

Figure 1

(a) Snapshots of the microfluidic platform from bottom and side views. (b) Schematic drawing with a coordinate system and relevant dimensions. (c, d) Time-averaged flow-induced birefringence (FIB) across the channel width for a representative test fluid composed of CNC at ϕ = 1 × 10–3 suspended in water at an average velocity U = 1.6 × 10–3 m s–1. In (c) the normalized birefringence intensity (Δn/ϕ) and in (d) the orientation of the slow optical axis (θ), given by the contour plot (in full resolution) and depicted by the solid segments to guide the eye. The horizontal dashed lines in (c) and (d) indicate the location where the spatially averaged birefringence, ⟨Δn⟩, and the spatially averaged angle of slow axis, ⟨θ⟩, are obtained for quantitative analysis.

Flow-Induced Birefringence (FIB)

Time-averaged FIB measurements were performed using an Exicor MicroImager (Hinds Instruments, Inc., OR) using a 5× objective at room temperature (∼22 °C). The channel is illuminated through the z-axis (vorticity direction) using a monochromatic beam of wavelength λ = 450 nm or λ = 630 nm (see Figure a,b). The retardance, R in nm, and the orientation of the slow optical axis (extraordinary ray), θ, were obtained from seven images acquired at 1 s intervals and time-averaged. The spatially resolved (spatial resolution of Δn and θ is ≈2 μm/pixel) and time-averaged birefringence (Δn = R/H) and the orientation of the slow optical axis, θ, are obtained in the flow-velocity gradient plane (x–y plane), as shown in Figure c,d, respectively, for a representative test fluid of dilute CNC in water. For CNC, θ probes the angle between the main CNC axis and the flow direction projected in the flow-velocity gradient plane. We note that the birefringence originates from the collective alignment of all the CNC rods present through the optical path (z-axis). At ϕ = 1 × 10–3 we estimate that a total of rods/pixel are probed simultaneously in the FIB experiment (see Supporting Information). As multiple rods are probed over a relatively long time frame (7 s), our FIB setup provides insights regarding the most favorable orientation of CNC under flow but not on the detailed orientation statistics such as the CNC orientation distribution. For this reason, we note that occasional tumbling of a small population of the CNC cannot be identified. For each flow rate, the birefringence Δn and the absolute value of the orientation of the slow optical axis, |θ|, are spatially averaged along 1 mm of the x-axis at y = ± 0.1 mm (see dashed lines in Figure c,d) and referred to as ⟨Δn⟩ and ⟨θ⟩, respectively. The background value of Δn acquired at rest was subtracted for all the analyses presented, and quantitative analysis of θ is restricted to Δn > 3 × 10–6. The orientation angle ⟨θ⟩ probes the CNC orientation with respect to the flow direction, whereas the birefringence intensity, ⟨Δn⟩, probes the extent of anisotropy in the system. For fully isotropically oriented particles, ⟨Δn⟩ = 0 with colloidal alignment occurring for ⟨θ⟩ < 45° and ⟨Δn⟩ > 0.[49,50] The error related to the spatially averaged ⟨Δn⟩ and ⟨θ⟩ is the standard deviation from the averaging process.

Rheology and Flow Simulations

Shear rheometry of the test fluids was performed using a strain-controlled ARES-G2 rotational rheometer (TA Instrument Inc.) equipped with a stainless steel cone and plate geometry (50 mm diameter and 1° cone angle). The test fluids were covered with a solvent trap and measured at 25 ± 0.1 °C (controlled by an advanced Peltier system, TA Instruments). The viscosity of the tested fluids was measured in “steady-state sensing” mode as a function of the shear rate . The rheometer determines automatically when the measurement reaches a steady state before moving to the next shear rate. For all fluids, the measured viscosity was independent of the preshear protocol with no hysteresis observed during either increasing or decreasing shear rate ramps, indicating negligible thixotropic effects. The data acquired below a torque limit of 0.5 μN m were discarded. The shear viscosity data were fitted to the Carreau–Yasuda (CY) generalized Newtonian model:where η0 is the zero-shear-rate viscosity, η is the infinite shear-rate viscosity, is the characteristic shear rate for the onset of shear thinning, n is the power law exponent, and a is a dimensionless fitting parameter that controls the transition to the shear-thinning region. The velocity field along the channel width (W, y-axis) and the value of the shear rate at y = |0.1| mm (the channel location where ⟨Δn⟩ and ⟨θ⟩ are obtained) were computed using numerical simulations. In the simulations, we consider the idealized problem in which an incompressible generalized Newtonian fluid with constant density ρ flows between two parallel infinite plates at a uniform temperature. The distance between the plates is W. The problem is to determine the velocity field together with the stress field. The velocity of the fluid is assumed to vary only in the y-direction; entrance and exit effects, as well as the presence of side walls, are neglected. Gravitational phenomena are not taken into consideration. Moreover, steady state is assumed and all time derivatives are neglected. The simplified equation of motion can be written aswhere P is the thermodynamic pressure and τ is the shear stress. According to the CY model,

Finally, the pressure gradient, ∂P/∂x, is calculated by imposing a specified mean velocity across the channel:The aforementioned system of equations is discretized and solved using an in-house Finite Element solver.[51] In all simulations, 200 linear elements were used across the width of the channel. For each average velocity used during the FIB experiment, ⟨Δn⟩ and ⟨θ⟩ are compared with the effective value of obtained from the simulation at y = |0.1| mm. The channel location of y = ±0.1 mm is chosen to be the midpoint between the side walls (y = ±0.2 mm) and the centerline (y = 0 mm) to provide relatively high values of shear rate while avoiding undesired wall effects in the FIB experiment. For very weak shear-thinning fluids, for which the CY model could not be fitted to the rheological data, the shear rate was computed like that for a Newtonian fluid.

Results and Discussion

Crowding-Free

We begin with the evaluation of the FIB of dilute CNC at ϕ = 1.0 × 10–3 suspended in aqueous Newtonian media, either in water (η = 0.9 mPa·s) or a water/glycerol mixture (17.2 vol % glycerol, ηs = 1.7 mPa·s), shown in Figure a–d. At this relatively low CNC concentration, interparticle interactions are negligible so that each CNC can be considered as isolated and crowding-free.[43]Figure a,c present the ⟨θ⟩ and ⟨Δn⟩/ϕ as a function of for the CNC in the crowding-free regime, respectively. For both media, the orientation angle, ⟨θ⟩, displays a gradual decrease with (Figure a), and for a given value of , the greater solvent viscosity of the water/glycerol mixture favors the CNC to align with a smaller value of ⟨θ⟩ compared to that for water as the solvent. Moreover, the greater viscosity of the water/glycerol media triggers the onset of CNC alignment at a lower shear rate than that in water, as indicated by the onset of birefringence occurring at lower values of (Figure c).

Figure 2

Spatially averaged orientation angle ⟨θ⟩ (a, b, e, f) and normalized birefringence ⟨Δn⟩/ϕ (c, d, g, h) for CNC suspensions in the crowding-free (a–d) and self-crowding (e–h) regime. Open and green-filled stars represent CNC in water and in a water/glycerol mixture (17.2 vol % glycerol), respectively (crowding-free). For the samples prepared in water, the CNC rotational diffusion coefficient in the crowding-free regime is = 59 s–1, whereas = 31 s–1 for the sample prepared in the water/glycerol mixture labeled by the green star symbol and used as a reference sample (Ref.). For the crowding-free regime, ⟨θ⟩ (a) and ⟨Δn⟩/ϕ (c) as a function of and scaled as in (b) and (d), respectively. For the self-crowding regime, ⟨θ⟩ (e) and ⟨Δn⟩/ϕ (g) as a function of Pe0 and scaled as in (f) and (h), respectively. The dashed lines in (a, b, e, f) are the fittings to eq with α = 0.39. In (b) eq is also plotted using α = 1 (solid line). Vertical lines at Pe0 = 1 and Pe = 1 are drawn as a reference. Error bars indicate the standard deviation of the measurement.

The effective rotational diffusion coefficient Dr of the CNC is obtained experimentally based onwhere 0 < α ≤ 1 is a stretching exponent that accounts for particle polydispersity.[50,52−54] For monodisperse particles, α = 1, whereas for polydisperse particles α < 1 (see solid and dashed lines in Figure b). Fitting the data in Figure a with eq yields α = 0.39 for both Newtonian solvents, whereas = 59 s–1 and = 31 s–1 for the water and water/glycerol solvents, respectively. The subscript “0” to Dr indicates the crowding-free regime (i.e., dilute CNC in a continuum medium) and is used to distinguish it from the case where Dr is affected by the surrounding crowding agent. By solving eq for l and using an effective CNC diameter d = 27.4 nm by accounting for the contribution from the electric double layer[41] and T = 22 °C, we obtain l = 610 nm, in accordance with the longest CNC population detected via microscopy in our previous work.[18] Our experimental data collapse onto master curves when we plot ⟨θ⟩ and ⟨Δn⟩/ϕ as a function of the respective Péclet number in the crowding-free regime, , and the birefringence signal increases sharply at Pe0 = 1, indicating that the Pe0 scaling is correct (Figure b,d). It is expected that the Pe0 scaling yields master plots of ⟨θ⟩ and ⟨Δn⟩/ϕ only for the systems where the CNC is not experiencing any confinement. It is important to note that ⟨Δn⟩ and ⟨θ⟩ are complementary parameters to quantify particle alignment during flow. The magnitude of birefringence ⟨Δn⟩ is linearly related to the number of aligned particles in a given illuminated volume, thus presented in a normalized form as ⟨Δn⟩/ϕ. Contrarily, for a uniform particle alignment, ⟨θ⟩ captures a geometrical property of the system that is independent from the number of aligned particles in a given illuminated volume.[49] With our experimental setup, we are unable to probe occasional tumbling of the CNC. However, because of the relatively high aspect ratio of the CNC (⟨l⟩/⟨d⟩ ≈ 55), we expect that deterministic tumbling does not occur within our experimental Pe0 range (0.01 < Pe0 < 300) and that the CNCs are mostly aligned in the flow direction at sufficiently high shear rates (details in the Supporting Information).[55]

Self-Crowding

Following the same approach as for the crowding-free case, we investigate the CNC alignment upon increasing CNC mass fraction ϕ so that analogous particles restrain the motion of each other, a regime that we refer to as self-crowding (Figure e,h). As a reference sample, we use the crowding-free suspension in water/glycerol mixture (see green star symbol). When the CNC concentration is increased, the values of ⟨θ⟩ decrease with increasing ϕ (Figure e). Analogously, the birefringence onset occurs at smaller values of Pe0, corresponding to lower values of (Figure g). This behavior can be explained by the restrained rotational motion of the CNC above the overlap concentration due to particle confinement, leading to a decreasing Dr with increasing ϕ. For each sample, Dr can be obtained by fitting the curves in Figure e with eq using a fixed value of α = 0.39, as established in the crowding-free regime. When ⟨θ⟩ and ⟨Δn⟩/ϕ are plotted as a function of the Péclet number , the curves collapse onto single master curves (Figure f,h) and eq leads to the dashed line in Figure f. We note that, for ⟨θ⟩, the master curve is set by considering only Pe as a scaling factor. Contrarily, for the birefringence ⟨Δn⟩, the CNC mass fraction ϕ needs to be considered; indeed, scaling the birefringence as ⟨Δn⟩/ϕ is required to collapse the curves onto a master curve.

Polymer Crowding

For the case of polymer crowding, we follow ⟨θ⟩ and ⟨Δn⟩ arising from the alignment of diluted CNC (ϕ = 1.0 × 10–3) in polymer solutions exposed to a shearing flow. We consider polymers with different Mw and concentrations to achieve viscoelastic polymer solutions with viscosity up to 400× the viscosity of water and relaxation times 0.04 < τp < 0.7 s to provide a large span of crowding environments to the CNC. We use aqueous solutions of poly(ethylene oxide) with Mw ≈ 4 MDa and Mw ≈ 8 MDa, referred to as PEO4 and PEO8, respectively, and polyacrylamide with Mw ≈ 5.5 MDa, referred to as PA5. In the absence of CNC, these polymer solutions do not display any significant birefringence, ensuring that ⟨Δn⟩ and ⟨θ⟩ signals measured for the CNC dispersions arise exclusively due to the CNC alignment. Here, we focus on PEO8 solutions; analysis for the PEO4 and PA5 solutions are given in the Supporting Information, Section 2. The confinement imposed on the CNC is tuned by the polymer concentration (c), expressed in normalized form as c/c*, where c* is the polymer overlap concentration (Figure ). For the PEO8, c* = 0.39 mg/mL. The PEO8 concentration in the polymer-crowded CNC dispersions is varied between the dilute regime, c/c* < 1, and the semidilute unentangled regime, where c/c* > 1 and the tube diameter (dt) is greater than the PEO8 radius of gyration, = 202 nm.[45] When the PEO8 concentration c/c* is increased, the ⟨θ⟩ decreases for a given shear rate; thus, (see Figure a), and the onset of birefringence shifts toward lower values of Pe0 (Figure c). Interestingly, when ⟨θ⟩ is plotted as a function of , using the values of Dr obtained from the fitting of eq , the curves collapse onto a master curve (Figure b), revealing that the alignment of the CNC occurs in a similar manner for different PEO8 concentrations. Elliptical hematite particles (with l = 600 nm and a cross section of 130 nm) in entangled PEO solutions have been reported to first orient along the flow direction and then evolve to orientations in the vorticity direction.[30] However, within our experimental window, we could observe only ⟨Δn⟩/ϕ increasing as a function of Pe, without the birefringence drop associated with the particle alignment drifting from the flow direction to the vorticity direction.[26,29,30] Indeed, perfect alignment along the vorticity direction (z-axis) for particles with circular cross sections would lead to isotropic projections in the flow–velocity gradient plane (x–y plane) with ⟨Δn⟩ = 0. Similar to us, Johnson et al.[26] observed only the orientation in the flow direction for elliptical hematite particles (with l = 360 nm and a cross section of 100 nm) in entangled polystyrene solutions.

Figure 3

Spatially averaged birefringence ⟨Δn⟩ and orientation angle ⟨θ⟩ for PEO8 solutions, at different c/c*, seeded with CNC at ϕ = 1.0 × 10–3. The PEO8 solutions are prepared in water; thus, the rotational diffusion coefficient in the crowding-free regime is = 59 s–1, whereas = 31 s–1 for the reference water/glycerol mixture, labeled by the green star symbol (Ref.). Orientation angle ⟨θ⟩ (a) and normalized birefringence ⟨Δn⟩/ϕ (c) as a function of and scaled as in (b) and (d), respectively. In (e) the birefringence ⟨Δn⟩ is scaled as ⟨Δn⟩/(ϕ – ϕiso) where ϕ – ϕiso = ϕeff and plotted as a function of Pe. (f) The ϕiso/ϕ (circles) and (dashed blue line) as a function of polymer concentration (c/c*). is the radius of gyration of the CNC and ξp is the polymer mesh size (eq ). The dashed line in (a) is an instance of the fittings to eq with α = 0.39. In (b) the curve described by eq collapses onto a master curve. Vertical lines at Pe0 = 1 and Pe = 1 are drawn as a reference. Error bars from panels (a)–(e) indicate the standard deviation of the measurement, whereas in (f) error bars indicate uncertainty associated with ϕiso calculation.

In contrast to the self-crowding cases, scaling ⟨Δn⟩/ϕ by Pe is insufficient to collapse the data onto a single master curve (Figure d). We re-analyze the data based on the hypothesis that the failure to collapse might be caused by a small population of CNCs which do not contribute to the ⟨Δn⟩ intensity by remaining within the PEO8 matrix. Indeed, scaling the ⟨Δn⟩ by an effective CNC mass fraction ϕeff = ϕ – ϕiso enables all the data to collapse onto a master curve (Figure e). For each c/c*, ϕeff is obtained by minimizing the sum of squared residuals between the reference water/glycerol curve and the polymer-containing samples (detailed procedure is given in the Supporting Information, Section 3). The ϕiso/ϕ increases with c/c*, and at c/c* ≈ 5, it seems to approach a plateau value ϕiso/ϕ ≈ 0.65 (Figure f). It is instructive to understand the evolution of ϕiso from a topological perspective. We compare the radius of gyration of the CNC, = 177 nm, using l = 610 nm as previously determined from , and d = 27.4 nm, with a mesh size of the PEO8[38]as a function of c/c*, where = 202 nm is the radius of gyration of PEO8. Specifically, we use the ratio to yield values between 0 (ξp → ∞) and 1 (ξp → 0). Both ϕiso/ϕ and the theoretical curve , based on eq , display a similar trend as a function of c/c*, suggesting that ϕiso is linked to the topological confinement exerted by the polymer mesh size ξp. Thus, at c/c* < 5, ξp is a strong function of the polymer concentration, and ϕiso/ϕ increases likewise. Contrarily, at c/c* > 5, ξp becomes a weaker function of c/c*; accordingly, ϕiso starts to plateau. This suggests that the topological confinement exerted by the polymer mesh size ξp on the CNC leads to a population of trapped CNC within the PEO8 network that is unable to align in the flow direction at Pe > 1; thus, it remains in the isotropic state. Nonetheless, with our current FIB setup, it is impossible to provide a definitive evaluation by which a small population of CNC remains in an isotropic state (ϕiso) within the PEO8 matrix. Molecular dynamic simulation could provide insights into the nature of the mechanism that leads to a population of CNC that is not affected by the flow. Overall, our observations imply that the PEO8 provides a two-way topological hindrance to the CNC rotation, where a fraction of the CNC, ϕeff, perceives the confinement but is able to align at Pe > 1, whereas the other remaining fraction, ϕiso, is unable to align in the range of investigated Pe. Contrarily, for the case of self-crowding, all the CNC contribute to the birefringence signals, that is, ϕ = ϕeff. This significant difference between self-crowding and polymer crowding can be associated with the network formed from flexible PEO8 polymer chaining versus the network composed of rigid rodlike CNC. In the following sections, we examine the dependence of ϕeff on relevant length and time scales.

Length-Scale Dependence

The rotational diffusion coefficient Dr obtained from eq captures the impact of the crowding environment on the CNC. To compare the Dr for the case of self-crowding and polymer crowding, we plot Dr against the statistical mesh size ξ of each crowding agent. The mesh size of the polymer matrix is given by ξp (eq ), whereas for the CNC suspension it is estimated aswhere ν is the number density of the CNC (number of CNC per unit volume).[17,56] Importantly, as ν ∝ l–1, ξr is independent from the rod length. For both cases, Dr decreases progressively with the decreasing ξ, indicating that the rods progressively sense the local confinement with the decreasing ξr or ξp. However, the CNC follows a much sharper decrease in Dr as a function of ξ for the case of self-crowding than that of the polymer crowding, indicating that the mesh size ξ alone is not able to fully capture the dependence of Dr from the crowding agent. The Dr trend for the self-crowding case shown in Figure a meets the expectation from rigid-rod theory, where in the dilute regime Dr is concentration-independent and = 1. While in the semidilute (self-crowding) regime, the CNC motion is constrained by the surrounding rods and Dr becomes concentration-dependent.[14] The dependence of Dr with the rod concentration has been described by the tube model in the framework of the Doi–Edwards theory, assuming that particles are rigid and monodisperse rods in the semidilute regime, aswhere β is a length-independent prefactor.[14,57,58] Recently, Lang et al.[17] experimentally validated for monodisperse colloidal rods that β = 1.3 × 103, as previously found from computer simulation.[59] Rearranging eq with eq , we obtainAlthough polydisperisity is not accounted for in the Doi–Edwards theory, it is interesting to note that the scaling captures the trend for the self-crowding case (Figure a). Moreover, when β = 1.3 × 103 and l = 610 nm are used in eq , it is possible to quantitatively capture the increasing with ξr in the semidilute (self-crowding) regime (see line in Figure a).

Figure 4

(a, b) Rotational diffusion coefficient of CNC, Dr, obtained from FIB (eq ), normalized by the respective rotational diffusion coefficient in the crowding-free regime, , as a function of the mesh size of the crowding agent ξ. (a) The case of self-crowding, given by an increasing CNC concentration with the corresponding mesh size ξr (eq ). (b) The case of polymer crowding, given by an increasing PEO8 concentration with the corresponding mesh size ξp (eq ). Relevant regimes for the (a) self-crowding and (b) polymer crowding are annexed above the respective panels. The line in (a) is the prediction from eq . In (b), the triangle indicates the scaling in the regime . (c) Comparison between the normalized zero shear viscosity measured by bulk shear rheology (empty symbols), and the viscosity experienced by the CNC, (filled symbols), obtained by eq . Error bars in (a, b) indicate uncertainty associated with Dr obtained from the fitting procedure of eq .

For the polymer-crowding case, we take inspiration from prior work on the translational and rotational diffusions of nanorods in PEO solutions.[20,22] Specifically, we adopt the scaling law by Cai et al.[20] developed for the translation diffusion coefficient of spherical particles in polymer solutions by considering as the characteristic dimension of CNC. This choice is motivated by the work of Alam and Mukhopadhyay,[22] which found that the scaling law developed by Cai et al.[20] satisfactorily predicted the scaling of rotational and translation diffusion coefficients of nanorods in polyethylene glycol solutions as a function of the polymer concentration. The majority of our data fall in the “intermediate regime”, where annotated above Figure b. In this regime, the scaling theory predicts , which captures our trend reasonably well.[20] We have also used other polymeric solutions (PEO4 and PA5) to gain further understanding of the polymer-crowding cases. We confirm that the mesh size alone is unable to fully describe the dependence of Dr from the crowding agent; thus, the curves of versus ξp do not collapse onto a master curve (see Supporting Information, Figure S3). Because CNC in the PEO8 solutions is in the dilute regime (ϕ = 1.0 × 10–3), it is instructive to evaluate the local viscosity experienced by the particles, ηlocal, as[23,24,60,61]We use = 59 s–1 determined in water with being the water viscosity and Dr obtained experimentally from eq (plotted in Figure a), to retrieve ηlocal. In Figure c, the zero shear-rate viscosity η0 obtained from bulk rheology (details in Figure b) is compared to ηlocal, displaying ηlocal < η0 in the investigated range of ξp. The mismatch between η0 and ηlocal indicates that the CNC does not perceive the surrounding medium as a continuum, thus experiencing a viscosity that lies between the water viscosity and the macroscopic bulk viscosity η0 of the polymer solutions. Consequently, predicting the minimum shear rate required to induce CNC alignment in polymer crowds, based on the criterion (i.e., Pe = 1), using the bulk viscosity η0 as the solvent viscosity in eq , would fail by underestimating Dr.

Figure 5

Steady shear viscosity measurements for (a) CNC suspended in water at different values of ϕ and (b) PEO8 solutions at different values of c/c* (without CNC) presented as versus . Solid lines in (a, b) describe the CY model (eq ). The onset of shear thinning, , obtained from the CY, and the value of Dr obtained from FIB measurements are plotted on the respective viscosity values. (c, d) Comparison between the Pe (filled symbols) and Wi (empty symbols) scaling for the normalized birefringence, ⟨Δn⟩/(ϕ – ϕiso), and orientation angle, ⟨θ⟩, for the CNC in PEO8 solutions at different values of c/c*. (e) Schematic of relevant time scales involved in the alignment of colloidal rods in polymer-crowding cases. The schematic is valid for polymers with relaxation time .

Time-Scale Dependence

Flow curves obtained from bulk shear rheometry were used to probe the two crowding agents, CNC suspensions in water and PEO8 solutions in the absence of CNC, under different shear rates and related flow time scales (i.e., ); see Figure a,b. It is important to note that the presence of CNC at ϕ = 1.0 × 10–3 in the PEO8 solutions did not alter the bulk rheology (see the Supporting Information, Figure S1). With increasing concentrations of the crowding agent (ϕ and c/c* for the CNC and PEO8, respectively), the shear viscosity increases and the onset of shear thinning, captured by from the CY model (eq ), shifts to lower values of shear rate (depicted by the yellow squares in Figure a,b). The Dr obtained via birefringence measurements (as shown in Figure a,b) are marked as triangular red symbols on the respective viscosity plots of the CNC suspensions and PEO8 solutions in Figure a,b.

For the CNC suspensions (Figure a), obtained from shear rheometry and Dr obtained via birefringence measurements have similar values because the onset of shear thinning correlates with the onset of CNC alignment. Therefore, the physical interpretation for the decreasing as a function of ϕ mirrors the one given for Dr. Specifically, in the semidilute (self-crowding) regime, the rods perceive the surrounding rods, enabling the onset of alignment at values of shear rate that decrease progressively with increasing ϕ. For the PEO8 solutions (Figure b), the onset of shear thinning, , corresponds to the longest relaxation time of the polymer solution as . In good approximation, the CNC alignment in PEO8 occurs at a shear rate (see yellow squares and red triangles in Figure b), suggesting that the onset of CNC alignment is coupled with the polymer relaxation time as . As such, a suitable control parameter for the CNC alignment is the Weissenberg number, , quantifying the strength of the elastic response of the fluid to an imposed deformation rate, where for Wi < 1 the polymers are in their equilibrium conformation but for Wi ≥ 1 the relatively high flow rate drives the polymers out of their equilibrium conformation.[62−64] On the basis of the relationship , the Weissenberg and Péclet numbers become identical because the rotational diffusion time scale of the CNC rods is the same as the longest relaxation time of the polymer for the regime investigated here (analogous results are also obtained for the PEO4 and PA5 solutions presented in the Supporting Information, Figure S4). Consistently, the trend of CNC alignment in PEO8 solutions captured by ⟨Δn⟩/ϕ and ⟨θ⟩ as a function of Pe is also well described by Wi, with the onset of CNC alignment occurring at Pe = Wi = 1 (Figure c,d). This is remarkably different from previous reports of elliptical hematite particles suspended in PEO solutions in the regime , where the onset of alignment occurs at Wi ≪ 1.[30] Similarly, the onset of alignment of carbon nanotubes in a sheared polymer melt was observed at Wi ≪ 1, ruling out the coupling of τp with Dr for relatively large colloids ().[29] From a topological perspective, the coupling between tracer particles and the polymer dynamics is predicted by Cai et al.[20] in the regime with the scaling (see Figure b). It is possible to conceptualize the coupling of Dr with τp by analyzing three distinct time scales at play during flow as sketched in Figure e. For , the polymer is relaxed and in its equilibrium configuration as the probed time scales are long enough to enable polymer relaxation, during which the CNC is able to escape from the transient confinement provided by the polymer mesh; thus, Brownian diffusion dominates. Increasing the shear rate, we reach , where the polymer is driven out of its equilibrium conformation and deformed by the flow. At this time scale the CNC perceives the surrounding polymer as a static mesh that provides confinement and aids CNC alignment at values of . At higher values of shear rate, , the CNC continues to align with the flow, following a universal curve of ⟨θ⟩ and ⟨Δn⟩/ϕeff with respect to Pe for a wide range of polymer concentrations in the semidilute unentangled regime, as shown by the master curves in Figure b,e. We note that the CNC alignment for the polymer-crowding and self-crowding cases is analogous. In Figure we plot together ⟨θ⟩ and ⟨Δn⟩/ϕeff as obtained for all the polymer- and self-crowding cases presented above. The CNC alignment can be described by the same master curve of ⟨θ⟩ and ⟨Δn⟩/ϕeff versus Pe. This observation implies that both self-crowding and polymer crowding of the CNC alters only the critical shear rate for the onset of alignment, but the subsequent trend in alignment for Pe > 1 remains the same for both cases. This universal trend with respect to the crowding agent is likely caused by the inability of the CNC to explore the surrounding confinement once the alignment is triggered by a sufficiently high shear rate (i.e., Pe = 1).[17] Note that in our present study the characteristic polymer time scale is greater than the rotational diffusion time scale of the CNC in the crowding-free regime, ; see the vertical line in Figure b with respect to . However, colloidal rods with slower rotational dynamics in the crowding-free regime compared to those of the characteristic polymer time scale will be in the regime . Practically, this regime can be achieved by increasing the length of the colloidal rods and/or decreasing the polymer molecular weight. As the regime is approached, by for instance increasing l, the colloidal rods will progressively experience the surrounding environment as a continuum rather than a discrete medium and ηlocal → η0. Therefore, for , the onset of colloidal alignment is expected to be dominated by the bulk viscosity of the surrounding polymer solution.[19]

Figure 6

(a) Orientation angle, ⟨θ⟩, and normalized birefringence, ⟨Δn⟩/ϕeff, as a function of Pe for the self-crowding (from Figure f,h) and polymer crowding (from Figure b,e) for a total of 16 data sets including the reference sample. The dashed line in (a) is the plot of eq using α = 0.39.

Conclusions

We tackle an industrially relevant problem from a fundamental perspective: the control over the alignment of rodlike colloids in polymeric matrixes. Specifically, we compare the flow-induced alignment of rigid colloidal rods, namely, CNC, in two contrasting crowded environments referred to as self-crowding and polymer crowding. By analysis of the length and time scales, we find that rotational diffusion coefficient, Dr, of CNC in high-molecular-weight polymeric crowds is coupled with the longest relaxation time of the surrounding polymer, τp. On this basis, we propose the Weissenberg number Wi as the control parameter for the alignment of colloidal rods that possess similar length scales as the suspending polymer fluid, , that is, in conditions where the continuum approach breaks down. Specifically, we show that by knowing τp from rheological tests, it is possible to predict the critical shear rate for the onset of colloidal alignment in polymeric fluids as ; equivalently, Wi = 1, without relying on the knowledge of the local viscosity experienced by the colloidal rods, ηlocal. In this work, we do not consider how the CNC dynamics are influenced by the polymer-depleted layer near the CNC surface because of the difficulty in estimating the depletion layer thickness for rodlike particles. We envisage that future work will be needed to understand the role of the depleted polymer layer, and its dynamic properties, on the CNC alignment under flow. In conclusion, our results provide crucial insights on the dynamics of colloidal rods under shearing flows that will aid the production of composite materials with desired structural organization. Additionally, the ability of tracer colloidal rods to probe the relaxation times of the surrounding polymers opens the opportunity to perform in situ and spatially resolved characterization of the dynamics of polymeric fluids under flow using tracer colloidal rods. With further optimization (e.g., size and composition of the colloidal rods), this technique is promising for analyzing polymer dynamics in complex flows encountered in real-life conditions where the investigation is a significant challenge. As a natural next step to our work, we envisage future studies where self-crowding and polymer crowding are at play jointly, mirroring actual industrial conditions. We use CNC as industrially relevant colloidal rods, but the basic principles will also apply to other anisotropic, rodlike, colloidal particles.