# Shearmetry in fluids by using size-controlled anisotropic LaPO 4:Eu 3+ nanorods as polarized luminescent probes

Qilin Zou, Zijun Wang, Lilian Magermans, Marcello Bonetti, Thierry Gacoin, Jongwook Kim

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Qilin Zou, Zijun Wang, Lilian Magermans, Marcello Bonetti, Thierry Gacoin, et al.. Shearmetry in fluids by using size-controlled anisotropic LaPO 4:Eu 3+ nanorods as polarized luminescent probes. Advanced Functional Materials, In press, 10.1002/adfm.202423439 . hal-05012366

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# Shearmetry in fluids by using size-controlled anisotropic LaPO₄:Eu³⁺ nanorods as polarized luminescent probes

Qilin Zou, Zijun Wang, Lilian Magermans, Marcello Bonetti, Thierry Gacoin* and
Jongwook Kim*

Q. Zou, Z. Wang, L. Magermans, M. Bonetti, T. Gacoin, J. Kim.

Laboratoire de Physique de la Matière Condensée, Ecole Polytechnique, CNRS, IP Paris,
91128 Palaiseau, France.

E-mails: jong-wook.kim@polytechnique.edu and thierry.gacoin@polytechnique.edu

**Keywords:** lanthanide, nanocrystal, polarization, microfluidics, shear stress

## Abstract

Orientation analysis of colloidally dispersed nanorods (NRs) provides a powerful means to measure local shear stress in fluids. Lanthanide-doped crystalline NRs, with their polarized luminescence properties, offer a promising platform for such *shearmetry* applications, as their emission spectra reflect the degree of collective orientation induced by shear. However, precise control over the morphology of the NRs is essential for ensuring reliable measurements, particularly in complex and dynamic flow environment. Herein, we present size-controlled synthesis of Eu-doped LaPO₄ NRs and their application in shearmetry within a microfluidic channel designed to generate curved streamlines. By analyzing 2D shear maps obtained using NRs of varied sizes, we quantitatively evaluate the impact of NR size on the shearmetry performance. Size-control of the NRs was achieved by adding a surfactant during hydrothermal synthesis. The collective orientation of NRs, induced by flow shear, was determined by analyzing the polarized Eu³⁺ emission spectra. We reveal that NRs of smaller aspect ratio (AR) and length respond more rapidly to shear variations, yielding symmetric and accurate shear stress profiles even in complex, dynamic flows. Additionally, smaller NRs extend the measurable shear stress range up to 0~15000 mPa, covering values typical of biofluidic systems and microfluidic devices. These results provide a novel framework for advanced fluidic analysis highlighting the critical role of the nanoprobe morphology.
# 1. Introduction

Phosphors with anisotropic structures generate polarized luminescence providing orientational sensitivity.[1] Emitters such as semiconductor nanorods (NRs), metal rods, or single anisotropic molecules have often been used for orientational analysis.[2] However, precise angular measurement has often been limited by the particle size-dependent polarization behavior or the unstable emission intensity due to photobleaching or blinking.[2d, 3]

Lanthanide-doped nanocrystals, on the contrary, exhibit a reliable feature of emission polarization that is independent of emitter size and morphology associated with high photostability.[4] The luminescence of lanthanide ions ($Ln^{3+}$) that are doped in crystalline host matrix consists of sharp and multiple emission lines originating from the crystal splitting of intra 4f-4f electron transitions.[5] The polarization of each sublevel transition is determined by the selection rules depending on the $Ln^{3+}$ site symmetry.[6] As a result, all sublevel emissions are polarized along particular directions with respect to the crystal axis of the emitter particle. Furthermore, the co-existence of electric and magnetic dipole transitions (ED and MD) allows 3-dimensional (3D) orientational analysis of not only single particles but also their collective assemblies.[7] Such a unique polarization feature relying solely on the intrinsic crystal structure has been reported in diverse upconversion or downshifting systems such as Yb/Er-doped $NaYF_4$, $LiLuF_4$ and $LiYF_4$, and Eu-doped $NaYF_4$ and $LaPO_4$ at a level of single microcrystals[7a, 8] or assembled nanocrystals[6b, 9], showing potential applications in single particle tracking, information transportation and microfluidics.[1c, 5a, 10]

By applying the collective orientational analysis of Eu-doped $LaPO_4$ NRs in flowing colloidal suspensions, we have recently demonstrated a novel shearmetry method to measure the local shear stress in microfluidic systems.[7b, 11] Compared to the commonly used methods like particle imaging velocimetry (PIV)[12] and particle tracking velocimetry (PTV)[13], which analyze velocity to estimate shear stress, our direct stress-optics measurements offer the advantages of local monitoring, high spatial resolution, and enhanced accuracy. The flow shear stress is directly correlated with the shear-induced orientational director $\vec{n}$ and order parameter $f$ of the NRs that can be retrieved from their polarized emission spectra. This unprecedented shearmetry technique allows us to study various micro- and bio-fluidic systems to unravel fluid dynamic and mechanobiological processes. However, it was found that the previously used NRs with large size (length > 200 nm, width > 10 nm) cannot instantaneously respond to the variation of shear stress, which limits the application of shearmetry in complex micro-channel geometries or in dynamic (non-stationary) flow systems. Peclet number (Pe), characterizing the relative rates of advection and rotational diffusion of the NRs, describes
how fastly the NRs respond to the variation of shear stress[7b]. Large Pe indicates a significant
delay in the rotation of NRs compared to their translation, causing the observed orientation of
NRs to reflect not only the current shear but also the past shear, namely a history effect. This
deteriorates the accuracy and sensitivity of the shearmetry. Measuring a reliable shear stress
profile in complex and dynamic fluidic systems, that are of high interest today, essentially
requires to improve the fundamental colloidal properties of the NRs to achieve their rapid
response to shear by minimizing Pe and symmetric shear profiles on a constriction.

In this work, we achieve control of the Pe value by adjusting the size of NRs. Theoretically,
small variations in nanorod size can greatly influence Pe and enhance shearmetry
performance. According to previous simulations[7b], Pe value below 0.2 can provide reliable
shearmetry results under most experimental microfluidic conditions. Here, we demonstrate
how variations in NR size alter the calculated shear stress profile, providing experimental
evidence that lower Pe values enable more accurate shear stress measurement, in agreement
with the theoretical prediction.

For this purpose, we investigate size-controlled synthesis of LaPO₄:Eu³⁺ NRs. LaPO₄ is an excellent host crystal for Eu³⁺ luminescence exhibiting high degree of polarization (DOP) and distinguished ED and MD emissions.[⁶b, ¹⁴] Nanosized LaPO₄:Eu³⁺ rods can be synthesized by hydrothermal methods.[¹⁴] And their outstanding colloidal stability, suitable for shearmetry applications, can be achieved in various media.[⁷b, ¹¹, ¹⁴⁻¹⁵] Nevertheless, the previous synthesis methods generally produce large-sized NRs leading to large Pe. To properly adjust the NR size, we employ a surface complexing agent (ethylenediaminetetraacetic acid disodium salt, EDTA) that coordinates with Ln³⁺ during the crystals' nucleation and growth processes. The average length and width of the NRs could be decreased down to 37 nm and 8 nm leading to an order of magnitude smaller Pe. Using these size-controlled NRs, we demonstrate Pe-dependent shearmetry mappings in a microfluidic channel with a geometrical constriction, and a largely extended sensitivity range of the shearmetry thanks to the adjustable size of the NRs.

**2. Results and discussion**

**2.1. Synthesis and characterizations of LaPO₄:Eu³⁺ NRs**

LaPO₄ can be crystallized in a mixture solution of La(NO₃)₃ and (NH₄)₂HPO₄ under mild
hydrothermal conditions. Following the previously developed microwave-assisted
hydrothermal strategy[14], Eu-doped LaPO₄ NRs were synthesized with further modifications. To precisely control the dimension of NRs, ethylenedinitrilotetraacetic acid (EDTA) disodium salt was added as the surfactant in the synthesis. The transmission electron microscopy (TEM) images in Figure 1a show the morphology change of as-synthesized LaPO₄:20%Eu NRs under different EDTA:Ln³⁺ ratios. One can clearly see that NRs become shorter while increasing the amount of EDTA. The size length (l), width (w), and aspect ratio (AR= l/w) of the as-synthesized NRs were measured on TEM images counting 100 NRs per sample and summarized in Table S1. The mean values of l and AR of LaPO₄:20%Eu NRs as a function of EDTA:Ln³⁺ ratio were plotted in Figure 1b and the corresponding length distributions were displayed in Figure S1. It is noted that the longitudinal growth of NR is more sensitive than transverse growth to the addition of EDTA. With the EDTA:Ln³⁺ ratio increases from 0.0002 to 0.1, the mean length is reduced from 182 ± 65 to 37 ± 9 nm while the mean width was slightly reduced from 10 ± 2 to 8 ± 1 nm. As a result, the mean AR was rapidly decreased from 18 ± 7 to 5 ± 1 nm. In comparison, the absence of EDTA generates large sizes of 195 × 7 nm and AR of 28 (Figure S2) under the identical conditions.

EDTA is a strong chelating agent that can bind to Ln³⁺ to form Ln-EDTA complexes. That chelation should bring out a more favorable particle nucleation and limit particle growth, resulting in smaller particle size according to the classical LaMer model[16]. This effect on size is in a good agreement with previous reports in other systems (NaYF₄[17] and BaYF₅[18]), by varying EDTA:Ln³⁺ mole ratio. High resolution (HR) TEM images and corresponding fast Fourier transform (FFT) images (Figure 1a, bottom right panels) display clear lattice fringes for NRs with largest and smallest ARs (18 and 5 respectively), indicating high crystallinity of obtained samples. For NRs with AR=18, the d spacing of 0.314 nm was measured and identified for (120) plane of monazite LaPO₄ in monoclinic structure, while for NRs with AR=5, the d spacing of 0.605 was identified for typical (100) plane of rhabdophane LaPO₄ in hexagonal structure. This implies that not only the size but also the crystal structure is affected by the addition of EDTA.

Fourier Transform Infrared (FTIR) spectra (Figure S3) demonstrate the existence of residual EDTA (1448 and 3032 cm⁻¹: C-H stretching vibrations of EDTA) only when the initial addition of EDTA was large enough, such as EDTA:Ln³⁺ of 0.1. The X-ray diffraction (XRD) result (Figure 1c) confirms the formation of pure monazite phase of LaPO₄ for EDTA:Ln³⁺ of 0.0002 (AR= 18) and the pure rhabdophane phase with broadened peaks due to reduced crystallite size (EDTA:Ln³⁺= 0.1, AR=5). When EDTA:Ln³⁺ ratio is increased 5 times to 0.001, the diffraction peaks already exhibit rhabdophane phase. The previous study
showed that high Eu concentration could also lead to phase evolution from monazite to rhabdophane.[14] Then, 5%Eu doped LaPO₄ samples were prepared with the same EDTA:Ln³⁺ ratios as for 20%Eu, aiming for only monazite phase. Nevertheless, Figure S4 still shows that the monazite diffraction peaks of LaPO₄:5%Eu gradually decrease, and rhabdophane peaks appear and become dominant, while increasing the amount of EDTA.

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Characteristics of as-synthesized LaPO₄:20%Eu NRs. (a) Low resolution TEM images of samples synthesized with different EDTA:Ln³⁺ mole ratios and the high resolution TEM images of selected samples with AR= 18 and 5, respectively. (inset photos: FFT patterns). (b) Estimated size and AR evolution as a function of EDTA:Ln³⁺ ratio. The average length and width of the anisotropic NRs were measured by counting at least 100 nanoparticles</figcaption></figure>
for each sample, then the ARs were further calculated. (c) XRD patterns of the as-synthesized samples (PDF#04-0635 (blue solid line): rhabdophane phase; PDF#84-0600 (red solid line: monazite phase). (d) Photoluminescence spectra for the samples under irradiation of 394 nm laser. (e) Hydrodynamic diameter of NRs measured by DLS in pH 2 aqueous suspension.

Based on above observations, we clearly see that EDTA has a significant impact on the phase transition of LaPO₄:Eu. It is known that rhabdophane is a kinetic phase that can be transformed into thermodynamically stable monazite phase.[6b, 14] The phase transition requires high activation energy and it is described as thermally induced dissolution/reprecipitation process.[14] In our synthesis, the added EDTA seems to anchor on the surface of NRs and reduce surface energy lowering the activation energy of the system. As a consequence, the rhabdophane could not be dissolved and recrystallize into monazite.

To evaluate the optical property of LaPO₄:20%Eu NRs, the emission spectra were recorded. Figure 1d displays the photoluminescence spectra of the LaPO₄:20%Eu NRs under laser excitation ($\lambda_{ex}$=394 nm). The emission peaks at 587, 592, and 595 nm are ascribed to $ ^{5}D_{0} \rightarrow ^{7}F_{1} $ magnetic dipole transition; the electric dipole transitions are split into corresponding peaks at 611, 615, 620 nm ($ ^{5}D_{0} \rightarrow ^{7}F_{2}$); 652 nm ($ ^{5}D_{0} \rightarrow ^{7}F_{3}$); 684, 688, 694, and 698 nm ($ ^{5}D_{0} \rightarrow ^{7}F_{4}$). The monazite sample with AR=18 apparently shows higher and sharper splitting peaks than that of other four samples in rhabdophane phase with smaller ARs. The rhabdophane NRs exhibit almost identical luminescence intensity regardless of the AR values. It is consistent with our previous study that Eu³⁺ doped LaPO₄ monazite has higher luminescence efficiency than rhabdophane.[6b] Eu³⁺ is an optical probe that can detect changes in local crystal structure[6b, 19] and site symmetry[20], because its optical transitions (luminescent intensity and peak ratios) vary with the surrounding coordination environment[21]. The normalized photoluminescence spectra (Figure S5) show two sets of Eu³⁺ emission profiles due to the different occupations of Eu³⁺ emitters in LaPO₄ crystals (D₂ symmetry in monazite for AR 18; C₁ symmetry in rhabdophane for ARs 13, 9, 8, and 5).

The as-prepared NRs show good colloidal stability in pH 2 water, which is confirmed by the measured small dynamic light scattering (DLS) sizes (Figure 1e and S6). These NRs were positively charged by high zeta potentials (> 30 mV, Figure S6), except for the sample AR 5 (with a zeta potential of 11 mV) made by adding the most EDTA amount.

In order to investigate the size effect on the microfluidic shearmetry in ethylene glycol (EG), a series of samples with a wide size range (selected ARs of 24, 18, 13, 8, and 5, the TEM image and photoluminescence spectrum of AR 24 can be found in Figure S7) were transferred
into EG which may act as capping shell and provide additional stabilization[7b, 22]. This allows better comparison with our previous work[7b, 11] that has been made in this media. The excellent dispersion stability of the NRs in EG is demonstrated by the SEM images in Figure S8, which guarantees the liquid crystal (LC) formation driven by self-assembly in anisotropic suspensions. Figure S9 shows the typical birefringence texture, verifying the formation of LC where the NRs were well-aligned in nematic domains. The phase separation was hardly to obtain for small ARs (8 and 5) because of the decreased steric hindrance induced by their shortened lengths and narrowed size distributions. Additionally, the DLS results (Figure S10) of NRs suspensions show that NRs are very stable in EG (VF 1.2%) without aggregations after aging 10 months under ambient conditions.

## 2.2 Reference polarization photoluminescence of LaPO₄:Eu

The prerequisite for precise orientation analysis is to acquire the reference spectra of polarized photoluminescence which should be obtained with NRs that are aligned all together towards a single direction. This can be realized by applying external electric field on the confined nematic domains between two electrodes, as illustrated in Figure 2a. As the selected samples were in either monazite or rhabdophane, the polarized photoluminescence spectra of two nematic domains from AR 18 (monazite) and AR 13 (rhabdophane) were respectively measured under optimal electric fields.
<figure><img src="image_3.png" /><figcaption>**Figure 2.** Characterizations of polarized photoluminescence of well aligned LaPO₄:Eu NRs under electric field. (a) Schematic illustration of NRs alignment via applying a homemade electro-optical cell. (b, d) Polarized photoluminescence spectra (recorded by a confocal microscope) from well-aligned monazite (AR 18) and rhabdophane (AR 13) nematic domains. A continuous 394 nm laser excitation with a power of 10 mW was adopted for the measurements. The spectra *I*<sub>π</sub> and *I*<sub>σ</sub> are representing photoluminescence of the aligned NRs parallel and perpendicular to polarizer respectively. Polar diagrams of selected electric and magnetic transitions, and corresponding transition ratios for (c) monazite and (e) rhabdophane. (f) The calculated DOPs of all presented sublevel transitions for both monazite and rhabdophane.</figcaption></figure>

Figure 2b shows polarization luminescence spectra of the monazite NRs measured at polarizer parallel (0°) and (90°) perpendicular to the NRs orientation, which can be referred as π and σ spectra respectively as the crystallographic c axis is parallel to the longitudinal direction of the rod. The polar diagrams for different peak intensities are plotted in Figure 2c, where the experimental points fit well with the symmetric dumbbell shapes trigonometric functions (I = A sin²θ + B cos²θ, where I and θ represents the integrated luminescence intensity and the angle between rod orientation and analyzer, respectively). One can see that the degrees of polarization for adjacent sublevel peaks are distinct to each other for both ED and MD transitions. For instance, the polar diagrams of MD1 and MD2 transitions are
orthogonal to each other as well as for those of ED1 and ED2 transitions (Figure 2c). It was demonstrated in our previous report that a ratiometric analysis of two transitions (MD1/MD2 or ED1/ED2) can precisely determine the orientation of NRs in three-dimension.[7] The accuracy of this spectroscopic method for geometrical orientation analysis is superior to other approaches based on non-ratiometric polarization optics.[7]

The polarized luminescence spectra and corresponding polar diagrams of the rhabdophane NRs are plotted in Figure 2d-e. While their global tendency of polarization is similar to those of monazite NRs, the spectral line shapes are obviously different. The rhabdophane shows slightly broader Eu³⁺ emission peaks maybe due to more structural defects formed compared to that of monazite, which is also in a good agreement with our previous observations comparing the polarized photoluminescence features from the different crystal phases.[6b] Regarding the magnitude of polarization quantitatively assessed by the degree of polarization (DOP) defined as $(I_π - I_σ) / (I_π + I_σ)$ (Figure 2f), the monazite always has a higher DOP than the rhabdophane for each peak as already reported for larger particle size[6b, 14]. The highest DOPs of 0.82 and 0.64 at 684 nm were obtained for monazite and rhabdophane respectively. Therefore, the accuracy of orientation analysis demonstrated below is generally higher for monazite samples.

## 2.3 Orientation analysis in microfluidics

A rod orientation is determined by the polar(θ) and azimuthal (φ) angles (Figure S11). When many rods are submitted to an external field such as electric field or flow shear, the preferential partial orientation (denoted as director $\vec{n}$) of the ensemble of rods obeys the applied electric field or flow direction[7b, 23]. Here, we consider a specific case of NRs flowing in the horizontal mid-plane of a straight microfluidic channel with a square cross section (Figure 3a), where $\vec{n}$ lies in-plane due to the vertical symmetry, and thus $\langle \sin^2 \theta \rangle$ and $\langle \cos^2 \phi \rangle$ are simplified to be 1/2. Therefore, for MD emissions, two polarized luminescence intensities parallel ($I_‖$) and perpendicular ($I_⊥$) to $\vec{n}$ can be expressed by[7b]:

$$I_‖ = \frac{I_σ}{2} (1 - \langle \cos^2 \theta \rangle) + \frac{I_π}{2} (1 + \langle \cos^2 \theta \rangle) \quad (1)$$

$$I_⊥ = I_σ \langle \cos^2 \theta \rangle + I_π (1 - \langle \cos^2 \theta \rangle) \quad (2)$$

where $I_π$ and $I_σ$ were obtained from perfectly aligned liquid crystal sample (Figure 3b). We choose two highly polarized MD transitions (MD1 and MD2) and their intensity ratio for
orientation analysis. Then, $\langle \cos^2 \theta \rangle$ can be deduced from Equation 1 by using the intensity ratio (such as $k_{\parallel} = I_{1\parallel}/I_{2\parallel}$):

$$ \langle \cos^2 \theta \rangle = \frac{k_{\parallel} \cdot (I_{2\sigma} + I_{2\pi}) - (I_{1\sigma} + I_{1\pi})}{k_{\parallel} \cdot (I_{2\sigma} - I_{2\pi}) - (I_{1\sigma} - I_{1\pi})} \quad (3) $$

The orientational order parameter defined by $f = (3\langle\cos^2\theta\rangle - 1)/2$ can then be calculated by measuring the polarization luminescence with polarizer at 0° (parallel to $\vec{n}$). The $f$ value is variable between 1 (perfect alignment along $\vec{n}$) and 0 (fully disordered). Figure 3a also shows the typical velocity and shear stress profiles on the cross section in a Poiseuille flow. The rods are highly orientated near the wall by a large shear stress, however, disordered in the center with ideal zero shear stress. Average shear rate can be calculated by $\dot{\gamma} = 6Q/wh^2$, where $Q$, $w$ and $h$ represent flow rate, width and height of channel respectively. Increasing flow rate, the same as increasing $\dot{\gamma}$, gives rise to a high $f$ value and changes the polarized luminescence spectrum toward the $\pi$ spectrum (Figure 3b). Hence, $f$ vs. $\dot{\gamma}$ correlation function can be established in this simple channel geometry, which can then be used to estimate unknown $\dot{\gamma}$ by analyzing polarized luminescence spectra collected from different local positions in various flow conditions.

<figure><img src="image_4.png" /><figcaption>**Figure 3.** Stress-optical measurements of NRs in microfluidics. (a) Illustration of flowing NRs in microfluidic channel. The profiles of shear stress ($\tau$) and velocity gradient ($\nu$) on the cross-section were plotted. (b) Polarization luminescence of NRs suspension in straight channel under different flow rate. (c) Established calibration curves of order parameter as a function of average shear rate for NRs with respective ARs. (d) Plotted evolutions of Exp. $\Omega$</figcaption></figure>
and Theor. Ω/Exp. Ω ratio with decreasing AR of NRs. (e) Estimated viscosity evolution as a function of shear rate for NR suspensions (Φ= 1.2%) with varying ARs. (f) The calibration curves of order parameter as a function of shear stress. The solid lines in Figure c, e and f were curve fitted. Error bars represent standard deviations from their corresponding multiple measurements.

Based on the methodologies described above, the *f* vs. *γ̇* correlation curves were obtained for the samples with different ARs (Figure 3c). Each series of data was obtained by nonlinear curve fits. *f* increases with continuously increasing *γ̇* and saturates over a certain level of *γ̇*. The *f* value for smaller AR is generally lower under the same *γ̇*, as smaller NRs are rotating faster and harder to be aligned[24]. Rotational diffusion coefficient (Ω) is a key parameter to quantitatively assess the rotational dynamics of NRs. Theoretically, Ω (Theor. Ω) is defined as:

$$ \Omega = \frac{3k_B T}{16\pi\eta_0 a^3} \left( -1 + 2\ln\left(\frac{2a}{b}\right) \right) $$

(4)

where *k*<sub>B</sub> and η<sub>0</sub> respectively represent the Boltzmann constant and the viscosity of solvent; *a* and *b* respectively represent half the length of the revolution axis and of the transverse axis of nanorod. The Theor. Ω values of NRs dispersed in EG (volume fraction = 1.2%) were calculated (Table S2) by using the measured sizes of NRs (Table S1), which shows a rapid increase, from 6 to 900 while the AR decreases from 24 to 5.

Meanwhile, the experimental Ω (Exp. Ω) values were evaluated by fitting the *f* vs. *γ̇* correlation curves with the numerical *f* vs. *γ̇/Ω* functions.[25] The fitting analyses for each different ARs are demonstrated in Figure S12 in which Ω was numerically estimated for each plot to achieve the best fit with the corresponding calibration curve (red line) from literature[25]. The obtained Exp. Ω values were listed in Table S2 and plotted in Figure 3d. The Exp. Ω also increases while AR decreases, but less dramatically than the Theor. Ω when approaching small AR. Thus, there ratio Theor. Ω/Exp. Ω rises up to 18 for the minimum AR=5 (Figure 3d). We attribute this discrepancy to the experimental colloidal properties; the size polydispersity and the interparticle interaction due to the surface charge. By dynamic light scattering (DLS) measurements, we found that the hydrodynamic size is significantly smaller than the individual rod lengths measured by TEM and zeta potential drops continuously when the AR decreases (Figure S6). Because our standard DLS analysis model
assumes spherical shape of particles, the calculated hydrodynamic size is usually smaller than the real length of anisotropic particles. Smaller zeta potential tends to increase Ω[7b], and larger size tends to decrease Ω more sensitively (Equation 4). The evolution of Exp. Ω would thus be a result of the balance between these two effects.

In order to determine shear stress ($\tau = \mu \cdot \dot{\gamma}$, where $\mu$ is dynamic viscosity) in non-Newtonian fluids where $\mu$ varies as a function of $\dot{\gamma}$, the correlation of $f$ vs. $\tau$ can be plotted once the $\mu$ ($\dot{\gamma}$) function is determined.[11] The methodology of determining dynamic viscosity of nanorod suspensions can be found in experimental section and Figure S13. Our NR suspensions in EG show shear-thinning behavior[26] in a small shear rate range (0~2 s⁻¹); $\mu$ drops at the initial stage when increasing $\dot{\gamma}$ (Figure 3e). Shear-thinning behavior is common for anisotropic colloids of which the significant interparticle interaction originating from geometrical perturbation decreases when the particles are aligned under flow[27],[28]. One can see that NRs with larger AR exhibit initially higher $\mu$ that drops quickly after shearing (inset in Figure 3e). The final $\mu$-value for all ARs is close to 16 mPa·s that is the viscosity of pure EG at 25 °C. The relatively lower $\mu$-value for AR=13 seems to be due to a slightly higher water content in this sample left during the transfer of NRs from water to EG.

Figure 3f shows the as-obtained $f$ vs. $\tau$ correlation curves. They show that larger AR provide higher accuracy (less error and better fit with the numerical function) for determining $\tau$ because $f$ saturates at higher level. However, the sensitive range of $\tau$ is extended by a factor 3 from AR=24 (0~5000 mPa) to AR=5 (0~15000 mPa). One can thus choose an optimal size of NRs depending on the range of $\tau$ applied in different fluidic systems, for instance, $\tau = 100\~9500$ mPa for human blood flow[29], $\tau = 0\~3$ mPa for ciliary beating in human airways[30].

## 2.4 Tomography of shear stress

By employing a scanning confocal fluorescence microscopy equipped with a spectrometer, we performed a local shearmetry with a submicron spatial resolution and its tomographic mapping. To assess the effect of particle size on the response time of NRs to the variation of shear, we used a microfluidic channel with a hemispherical constriction (Figure 4a). In this channel geometry, streamlines are curved surrounding the constriction, and the direction of shear varies accordingly. In principle, $\dot{\gamma}$ and $\tau$ are near zero at the center of channel section, and are maximum near the wall. Figure 4b represents the polarized emission spectra collected at the wall and center, which show the strong polarization (alignment due to high shear) at the wall and almost no polarization (disorder under zero shear) at the center.
<figure><img src="image_5.png" /><figcaption>**Figure 4.** Local shear stress tomography in microfluidics. (a) Schematic illustration of the microfluidic a semicircular constriction of diameter 50 µm used for local shear stress measurements. (b) Polarization luminescence spectra of MD transitions at the center and wall of the microfluidic channel. (c, d) The shear stress maps for AR 24, 18, 13, 8 under the identical flow rate of 0.02 µL/min. (e) The shear stress maps for AR 5 under different flow rates. All suspensions have the same volume fraction of 1.2% and the flow direction from left to right.</figcaption></figure>

Scanning the polarized luminescence on the horizontal mid-plane around the semicircular constriction (Figure 4a) enables us to construct the tomographic maps of the estimated order parameter (Figure S14), and τ (Figure 4c-e) by using the *f* vs. τ calibration curves for different ARs (Figure 3f). Apparently, particle size has a dramatic impact on the estimated τ profiles. Although a τ-map of a laminar flow in such a microscale should be strictly symmetric around a symmetric constriction, the τ-maps obtained with high ARs of 24 and 18 display a significant fore-aft asymmetry (Figure 4c-d). Such an asymmetric profile of estimated τ-map was observed in our earlier study, attributing it to the small Ω value representing slow rotational dynamics of NRs compared to the flow rate[7b]. And it was predicted that a correct symmetric τ-map would be achievable if Ω is increased by using smaller NRs. Figure 4d-e show an experimental proof of this prediction: The τ-map profile becomes more and more
symmetric when decreasing the AR. For small ARs of 8 and 5, the $\tau$-maps are almost perfectly symmetric with the region of maximum $\tau$ occurring at the top center of the constriction. Meanwhile, the $\tau$-map becomes again asymmetric when the flow rate is increased (Figure 4e).

These observations can be understood by comparing the timescales of advection ($t_a = H/U$, where $H$ is the channel width and $U$ the flow velocity) and diffusion ($t_b = \Omega^{-1}$). The Peclet numbers ($Pe = t_b/t_a = U/H\Omega$) for all cases of AR vs. flow rate in Figure 4c-e are provided in Table 1. $Pe$ decreases from 1.08 to 0.04 while AR decreases from 24 to 5, and increases again to 0.4 when flow rate increases. $Pe$ of 1.08 indicates that both advection and diffusion occur over equal times. The rotation of NRs is thus subject to a history effect that renders the NRs to slowly reorient to adapt the change of the flow around the constriction. Such a slow response results in the unreliable estimation of local $\tau$-value and the asymmetric $\tau$-map. When $Pe$ is considerably smaller than 1, the rotation of NRs is sufficiently faster than the advection allowing for the reliable $\tau$-mapping suppressing the history effect. The experimentally obtained $\tau$-map profiles for different $Pe$-values shown in Figure 4c-e are in a close agreement with the numerical simulations in our previous work$^{[7b]}$.

The above findings highlight the necessity of size optimizations for shearmetry technique. The history effect can be suppressed with small NRs. For the potential applications in different biofluid systems, we may suggest NRs with adaptable sensitivity ranges to meet the corresponding requirements. For example, the measured maximum shear stress of 5000 mPa for AR 5 under 0.2 $\mu$L/min is already reaching the saturation range of long NR, which demonstrates short NRs more suitable for high shear stress determination.

<table><caption>**Table 1.** A summary of Exp. $\Omega$, shear stress sensitive ranges, $Pe$ values for the suspensions under different flow rates, and profiles of shear stress tomography.</caption><thead><tr><th>AR</th><th>Exp. Ω</th><th>Shear range (mPa)</th><th>stress</th><th>Flow rate (μL/min)</th><th>Pe</th><th>Tomography</th></tr></thead><tbody><tr><td>24</td><td>1.71</td><td>0~5000</td><td>0.02</td><td>1.08</td><td>Asymmetric</td></tr><tr><td>18</td><td>12.54</td><td>0~10000</td><td>0.02</td><td>0.15</td><td>Asymmetric</td></tr><tr><td>13</td><td>29.63</td><td>0~12000</td><td>0.02</td><td>0.06</td><td>Asymmetric</td></tr><tr><td rowspan="2">8</td><td rowspan="2">44.30</td><td rowspan="2">0~14000</td><td>0.02</td><td>0.04</td><td>Symmetric</td></tr><tr><td>0.02</td><td>0.04</td><td>Symmetric</td></tr></tbody></table>
5

49.22

0~15000

0.1

0.2

0.2

0.4

Asymmetric

Asymmetric

### 3. Conclusion

In summary, we have studied size-dependent orientation analyses using polarized luminescent Eu-doped LaPO₄ NRs and their application for shearmetry in microfluidics. The excellent colloidal stability of the prepared NRs guarantees the high accuracy of spectroscopic characterizations. The local shear stress sensitivity is tunable in a wide range (0~15000 mPa) thanks to the adjustable size of NRs, promising versatile potential applications in various fluidic systems. The Pe-dependent shear stress tomographic mappings demonstrate that small NRs (low Pe) effectively suppresses the history effect on the orientation of NRs showing the correct symmetric shear stress profiles around a constriction. Small NRs are thus more favorable for shearmetry in non-stationary or curved flows while there exists a tradeoff with the shear sensitivity. These fundamental investigations of size-dependent orientation analyses of NRs and shearmetry pave a way to their practical applications in various microfluidic systems. And the capability to analyze dynamic shear stress with downsized NRs is particularly interesting for investigating complex biological microenvironments around diverse pathological tissue or cells, such as thrombus, cancer cells, and cilia.

### 4. Experimental Section

**Chemicals:** Lanthanum(III) nitrate hexahydrate La(NO₃)₃·6H₂O, europium(III) nitrate pentahydrate (Eu(NO₃)₃·5H₂O, 99.9%), ethylenedinitrilotetraacetic acid (EDTA) disodium salt, di-ammonium phosphate ((NH₄)₂HPO₄), ethylene glycol (EG) and nitric acid (70%, A.R.) were purchased from Sigma-Aldrich and used as received without further purification.

**Fabrication of microfluidic channels:** All the microfluidic channels (Y-shaped, straight, semicircular constricted) were fabricated based on corresponding molds. The polydimethylsiloxane (PDMS) channels were made by an organometallic crosslinking reaction. Briefly, a homogeneous mixture of SYLGARD 184 silicone elastomer base (30 g) and curing agent (3 g) was degassed and poured onto the channel molds. After further degassing, the PDMS was cured by heating at 65 °C for overnight. The PDMS with patterns were carefully cut and washed. After a plasma treatment (60 s) for cleaned glass slides and PDMS, they were attached together and put in the oven at 90 °C for overnight.
**Polarized luminescence measurements for establishments of calibration curves:** The measurements were conducted on the typical microscope (Olympus BX51WI, 20× objective, NA = 0.3) with inserted the linear polarizer. Nanorod suspension was pumped into the straight channel and the flow rate gradually increased. At each flow rate, polarized luminescence spectra were collected with the polarizer parallel (0°) and perpendicular (90°) to the flow direction. The dependence of average order parameters on shear rates were established by using the established reference spectra and the recorded polarized luminescence spectra.

**Shear stress tomography:** High-resolution scanning was implemented on a commercial confocal optical microscope. A long-pass dichroic mirror was used for reflecting excitation beam and transmitting emissions into the spectrometer. A polarizing beam-splitter was used to polarize photoluminescence. A 50 µm pinhole and a rotating half-wave plate were placed between the dichroic mirror and the polarizing beam-splitter. For the measurements, the NR suspension was pumped into the microfluidic channel of semicircular constriction to create a flow. A cross section dimension of 80 × 120 µm² (covering the constriction) was scanned by a step size of 1 µm for all the experiments. Three polarized luminescence spectra at each step were recorded with the polarizer at 0°, 45°, and 90°. 10 mW laser power and 50 ms exposure time were used to record each polarization photoluminescence spectrum. The orientation analysis was carried out by previously reported method.[7b, 11] The local shear stress was calculated by using the determined order parameter and the established *f* vs. *τ* calibration curves calibration curves.

**Polarization luminescence spectra analysis:** The Eu³⁺ emission peaks in the spectral region of 581-589.5 nm, 590-593 nm, 593.5-599 nm, 608-613.2 nm, 618-631 nm, 680-687.2 nm, 697.5-704 nm, 613.7-617.5 nm, 687.3-692 nm, 692-695.1 nm were integrated for polar diagram plotting, DOPs and order parameter calculations. Baselines of all those luminescence spectra were taken as flat line by averaging the intensity in the Eu³⁺ non-emission range at 640-644 nm.

**Characterizations:** Transmission electron microscopy (TEM) was performed using JEOL JEM-2010F (300 kV). Powder X-ray diffraction (XRD) measurements were performed on a Bruker D8 Advance diffractometer at Cu Kα radiation (λ = 1.5409 Å). Scanning electron microscopy (SEM) images were obtained on a Hitachi S4800 microscope (5 kV). Zeta potential and dynamic light scattering (DLS) size were measured by a Malvern ZetaSizer Nano ZS. A conventional microscope (Olympus BX51WI, 20× objective, NA = 0.3) equipped with a spectrometer (IsoPlane SCT-320, Teledyne Princeton Instruments), a CCD camera (PIXIS 400, Teledyne Princeton Instruments), a linear polarizer and a 394 nm continuous-
wave laser, was used for the measurements of classical photoluminescence (without inserting
the polarizer), polarized luminescence (for calibration curves), birefringence and viscosity.
The measurements of reference spectra and shear stress tomography were carried out on a
confocal microscope (Nikon Ti-DH) equipped with a dichroic mirror, an oil immersion
objective (60x, NA = 1.4), a 50 µm pinhole, a half-wave plate, a polarizing bea-splitter, a
spectrometer, a CCD camera and a piezo stage.[11]

**Supporting Information**

Supporting Information is available from the Wiley Online Library or from the author.

**Acknowledgements**

We acknowledge INSCOPER for providing the synchronized imaging solution. We are
grateful to Clément Marchat and Sandrine Tusseau-Nenez for help of SEM and XRD
measurements. This research was supported by Medical Research Foundation
(DCM20181039556, Chemistry for Medicine program) and French National Research
Agency (ANR-19-CE09-0033 and ANR-23-CE42-0007-01).

Received: ((will be filled in by the editorial staff))

Revised: ((will be filled in by the editorial staff))

Published online: ((will be filled in by the editorial staff))

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TOC

Size-controlled anisotropic LaPO₄:Eu³⁺ nanorods as polarized luminescent probes are exploited to achieve accurate shear stress tomography in complex dynamic fluids.

Qilin Zou, Zijun Wang, Lilian Magermans, Marcello Bonetti, Thierry Gacoin* and Jongwook Kim*

Shearmetry in fluids by using size-controlled anisotropic LaPO₄:Eu³⁺ nanorods as polarized luminescent probes

<figure><img src="image_6.png" /></figure>