We systematically investigated the impact of boundary confinement on the shear-thickening rheology of dense granular suspensions. Under highly confined conditions, dense suspensions were found to exhibit size-dependent or even rarely reported nonmonotonic ( $S$-shaped) flow curves in steady states. By performing *in situ* boundary stress microscopy measurements, we observed enhanced flow heterogeneities in confined suspensions, where concentrated high-stress domains propagated stably either along or against the shear direction. By comparing the boundary stress microscopy results with macroscopic flow responses, we revealed the connection between nonmonotonic rheology and stress heterogeneity in confined suspensions. These findings suggest the possibility of controlling suspension rheology by imposing different boundary confinements.

## I. INTRODUCTION

Dense granular suspensions comprise concentrated non-Brownian particles mixed with a Newtonian liquid. They are omnipresent in diverse natural phenomena and engineering applications, including mudflows in landslides [1], construction materials [2], and field-responsive fluids [3,4]. Unlike Newtonian fluids, the rheological response of a dense granular suspension to shear is often nonlinear. When suspension viscosity increases under shear, the flow behavior is referred to as shear thickening [5–7].

The physical mechanism of shear thickening can be understood across a hierarchy of length scales. Recent studies have interpreted shear thickening in dense non-Brownian suspensions as a shear-induced crossover from an unconstrained flow state to a constrained frictional state [8–10]. Within the shear-thickened flows, the suspended particles form direct contacts by overriding critical stress, depending on the intrinsic interactions among particles [11,12]. The continuous strengthening of large-scale contact networks results in a sharp increase in flow resistance [13,14]. This phenomenological framework not only captures the essential characteristics of shear thickening in dense suspensions [11] but also predicts the emergence of $S$-shaped flow curves above a critical volume fraction [8,15]. However, these nonmonotonic rheological responses have rarely been observed in steady state measurements [16]. Instead, unstable flow instabilities and substantial stress fluctuations often dominate the flow responses in this regime [17–20].

In addition to multiscale contact networks within dense suspensions, shear boundaries play essential roles in shear thickening. As dense suspensions tend to dilate under shear [21,22], the confining plates apply positive normal stresses to stabilize the contact networks, enabling sustained thickening responses [11,23]. Further, the geometric profiles of shear boundaries have been shown to control statistical features, including stress fluctuations [17] and flow heterogeneities [24–26], in dense suspensions. However, despite these experimental findings, the underlying mechanism by which the boundary conditions modulate the shear thickening flows remains unclear.

In this work, we systematically characterized the role of boundary confinement in both *sedimenting* and *nonsedimenting* granular suspensions. By combining rheological characterizations with boundary stress microscopy, we show that strong confinement imposed by shear boundaries effectively stabilizes nonmonotonic flow curves and induces stress heterogeneity. Driven by the interplay between shear boundaries and particle interactions, the high-stress domains propagate dynamically, either along or against the shear direction. These emergent features of confined suspensions are greatly reduced when not confined.

## II. MATERIALS

Athermal granular particles are commonly subjected to the influence of gravity. Therefore, particle-sedimentation plays an essential role in the rheological behaviors of many granular suspensions [17,21,27,28]. On the other hand, for particles having polymeric components, it is possible to match their densities to that of the solvent [11,12,26,29]. Nonsedimenting suspensions are usually stabilized by intrinsic inter-particle repulsions, such as electrostatic [30] and steric interactions [31]. In this study, we investigated the shear thickening rheology of both systems under confinement.

The nonsedimenting suspensions were prepared using polystyrene (PS) particles having a diameter of approximately $ d p s=25\mu $m, suspended in a glycerol-water mixture. The solvent made of 20 wt.% glycerol dissolved in 80 wt.% de-ionized water had a density of $ \rho g w=1.05 g/ cm 3$ and a viscosity of approximately 1.75 cSt. By measuring the settling speed of PS particles in the glycerol-water mixture, we estimated the density difference between the PS and the solvent to be as small as $\Delta \rho =2.2\xd7 10 \u2212 3 g / cm 3$. Since the resulting gravitational stress ( $\Delta \rho g d p s\u223c 10 \u2212 4$ Pa) was substantially smaller than the minimum shear stress applied by the rheometer, the effects of sedimentation were negligible in the PS-water suspensions. The PS particles can be well dispersed and stabilized by electrostatic repulsions in the suspensions [32]. To demonstrate this, we added sodium chloride (NaCl) to PS-glycerol suspensions. For a constant volume fraction of particles ( $\varphi =60%$), the yield stress rose as the salt concentration increased from 0.10 to 0.63 mol/l (Fig. S1 in the supplementary material), confirming the role of charge stabilization in PS-glycerol suspensions.

The sedimenting suspensions were prepared by mixing glass beads ( $ d g=43\mu $m) with 20 cSt silicone oil. Due to the density difference between glass ( $2.33 g / cm 3$) and silicone ( $0.95 g / cm 3$), the particles were consistently affected by gravity. Thus, the glass particles sedimented in static states, leaving a thin fluid layer between the particles and the shear plate. As the shear stress increased, the particles gradually lifted against gravity [17,21]. In this system, shear thickening occurred as the flow of particles spanned across the whole shear gap [22].

## III. SHEAR RHEOLOGY OF CONFINED SUSPENSIONS

### A. Steady-state rheological measurements

### B. PS-water suspensions

Figure 1(b) shows the flow curves of PS-water suspensions with $\varphi =0.60$, where the gap size was varied from $h=1.65$ to $0.50$ mm. For $h=1.65$ mm (open left-triangles) and $h=1.29$ mm (open circles), the suspension exhibited a Newtonian-like flow behavior, $\tau \u223c \gamma \u02d9$, below a critical stress $ \tau c 1\u22486$ Pa. For $\tau > \tau c 1$, $\tau ( \gamma \u02d9)$ displayed classical shear thickening responses, where $ \tau c 1$ is the stress required to establish direct contacts among PS particles by overcoming the electrostatic repulsion.

By further decreasing the gap size, we obtained size-dependent flow curves. For instance, the suspensions shear-thicken more profoundly under $h=0.86$ mm (solid circles), and even discontinuously at $h=0.56$ mm (solid up-triangles). More strikingly, a nonmonotonic flow curve of $\tau ( \gamma \u02d9)$ appeared as $h$ became as small as 0.50 mm (solid diamonds), where a flow regime with a negative slope, $ d \tau / d \gamma \u02d9<0$, emerges above $ \tau c$.

In addition, the increase in the viscosity of PS-water suspensions was associated with a synchronous rise of the normal stress ( $ \tau N$). As shown in Fig. 1(c), $ \tau N$ appears positive and increases with $\tau $ above $ \tau c 1\u22486$ Pa in all the measurements. For the suspensions with $h=0.56$ and 0.50 mm, containing approximately 20 layers of PS particles, the increase in $ \tau N$ due to thickening-induced dilation became appreciable. As demonstrated in the inset of Fig. 1(c), $ \tau N$ also exhibits a dependence on the gap size for $\tau > \tau c 1$. Given that $ \tau N$ scales almost linearly with $\tau $ under highly confined conditions (Fig. S2 in the supplementary material), we conjecture that the discontinuous and nonmonotonic flow curves in Fig. 1(b) emerge as the system size approaches the correlation length scale of frictional contact networks in the sheared PS-water suspensions.

### C. Glass-oil suspensions

We further characterized the rheology of sedimenting glass-oil suspensions as the gap size varied from $h=1.70$ to 0.68 mm. Due to the slow relaxation dynamics of the glass particles in viscous silicone oil, the shear rate $ \gamma \u02d9$ at each shear stress $\tau $ was obtained by averaging the instantaneous values over a period of $ t w=200$ s. As a further increase in $ t w$ did not vary the flow curves (Fig. S3 in the supplementary material), we conclude that the resulting $\tau ( \gamma \u02d9)$ represented the flow curves at steady states. Figure 1(d) shows $\tau ( \gamma \u02d9)$ of the glass-oil suspensions with $\varphi =52$% measured at different $h$. For each individual trace, we identified two distinctive Newtonian regimes at $\tau < \tau c 2=0.5$ Pa and $\tau > \tau e=6$ Pa, respectively. The crossover between the two flow regimes represents a shear thickening transition from a sedimenting to a suspended flow state. As $ \tau c 2$ is consistent with the gravitational stress $\Delta \rho g d g\u22480.58$ Pa where $\Delta \rho $ is the density difference between the glass and silicone oil, shear thickening in glass-oil suspensions was initiated by rearranging the particle configurations against sedimentation [17,21]. Similar to the PS-water suspensions, *S*-shaped flow curves for glass-oil suspensions emerged as the gap size was reduced to $\u223c20$ particle diameters: for $h=1.02$ and 0.68 mm, $\tau ( \gamma \u02d9)$ becomes nonmonotonic between $ \tau c 2$ and $ \tau e$, with $ \tau N$ rising more drastically than for large gaps [Fig. 1(e)].

### D. Stability of the S-shaped flow curves

As the shear stress $\tau $ was ramped up and down cyclically, we observed negligible hysteresis of $\tau ( \gamma \u02d9)$ in the PS-water suspensions [Fig. S4(a) in the supplementary material]. In contrast, for glass-oil suspensions under cyclic testing, the *S*-shaped traces still remained qualitatively unchanged, although the hysteresis loops emerged [Fig. S4(b) in the supplementary material]. These observations suggest that the nonmonotonic flow curves, which have been rarely observed in experiments [16], were stabilized by the imposed confinements. This mechanical stability is consistent with predictions in a previous simulation [15]: a confined suspension can potentially enhance particle contacts while reducing the shear rate, in response to an increasing shear stress.

We further examined the role of boundary roughness in the nonmonotonic responses. Using a shear plate roughened by a finer sandpaper (grain size $\u224850\mu $m), the $S$-shaped flow curves of PS-water suspensions disappeared at $h=0.50$ mm and then re-emerged as $h$ was further reduced to 0.33 mm [Fig. S5(a) in the supplementary material]. This observation suggests the essential role of boundary roughness in stabilizing nonmonotonic flows in the PS-water suspensions. With a reduced boundary roughness, wall slips can destabilize nonmonotonic flows [33]. In contrast, the *S*-shaped flow curves in confined glass-oil suspensions have a negligible dependence on the boundary roughness [Fig. S5(b) in the supplementary material]. As the glass particles are constantly affected by gravity, the shear plate is more likely to be in contact with the solvent film than with the particles, such that wall slips are prevented regardless of the boundary roughness.

## IV. LOCAL SHEAR STRESS FLUCTUATIONS

### A. Boundary stress microscopy

To search for the underlying origin of the nonmonotonic behaviors, we employed boundary stress microscopy (BSM) to characterize the shear-induced heterogeneity in confined suspensions. This technique was initially developed to measure in-plane tractions at soft interfaces [34,35] and was later improved for *in situ* measurements of boundary stresses in rheological characterizations [36,37]. Below, we briefly summarize the theoretical and experimental methods used in the BSM analysis.

The soft layers in our BSM setup were made of polydimethylsiloxane gels (PDMS from DMS-V31, Gelest Inc.) crosslinked by trimethylsiloxane copolymers (HMS-301, Gelest Inc.), with Poisson’s ratio $\nu =0.48$ [38]. The gel thickness was determined by the spin-coating speed applied to uncrosslinked PDMS mixtures, and Young’s modulus of a cured gel was controlled by the weight ratio of crosslinkers ( $k$). We deposited a layer of 5 $\mu $m fluorescent beads on the PDMS surfaces as tracers to measure $ u x \u2217$ and $ u y \u2217$ and to quantify the boundary stresses $ \sigma x z$ and $ \sigma y z$ using Eq. (8). Before each measurement, Young’s modulus of the gel substrate ( $E$) was determined by calibrating with a Newtonian fluid of known viscosity (Fig. S9 in the supplementary material). Since $E$ is independent of normal stress, the precision of the measured boundary stresses was not affected by the thickening-induced dilation. The objective was placed at a distance of $2R/3$ from the plate center, where the local shear rate was identical to the average shear rate reported by the rheometer [Eq. (2)]. The imaging speed was controlled between 3.8 and 7.6 frames per second, which was high enough to capture the evolving stress heterogeneity. As $ \sigma x z$ remained substantially larger than $ \sigma y z$ in all our experiments [Fig. 2(b) and Figs. S6 and S7 in the supplementary material], we focused only on the evolution of $ \sigma x z$ in this study.

To characterize the PS-water suspensions, we chose $k=0.97%$, $E=7.5$ kPa, and a gel thickness $ \delta 0=80\mu $m. During the shear thickening transition, the average shear stresses ( $\u223c 10 1$ Pa) were large enough to remove the nanobeads from gel surfaces. To prevent this, we added an additional PDMS film with a thickness of 6 $\mu $m on the top [Fig. 2(a)], such that the gel surface ( $\delta $) was slightly higher than the plane of nanobeads ( $ \delta 0$). We used $ \sigma i z$ $(i=x,y)$ at $z= \delta 0$ to estimate the boundary stress induced by the PS-water suspensions.

In contrast, the shear stresses in glass-oil suspensions were only $\u223c 10 0$ Pa, substantially lower than those of PS-water suspensions. We thus chose $k=0.81%$ and no additional PDMS film was needed above the nanobeads ( $\delta = \delta 0=90\mu $m). To prevent the swelling effects due to the oil component in solvents, we swelled the gel layers with 20 cSt silicone oil before the experiments. Consequently, the fully swollen PDMS layer has a thickness of 92 $\mu $m and Young’s modulus of $E=1.7$ kPa.

### B. Ps-water suspensions

In the BSM setup, the boundary roughness decreased due to the presence of a soft layer on the bottom plate. Consequently, although size-dependent flow curves of PS-water suspensions still emerged at $h\u22641.10$ mm, there were no nonmonotonic behaviors even at $h=0.31$ mm, as shown in Fig. 3(a). This result is consistent with our aforementioned observation that the rheology of PS-water suspensions is sensitive to boundary roughness. As the size-dependent rheology is the precursor of nonmonotonic flow curves (Fig. 1), we conjecture that the boundary stresses characterized in this regime ( $h\u22641.10$ mm) remain important indicators of the local flows in nonmonotonic regimes.

By maintaining a constant shear stress $ \tau a=20$ Pa in the thickening regime, we performed BSM measurements to characterize the evolution of local flows at $h=0.31$, 1.10, and 1.94 mm. Figure 3(c) shows that $\u27e8 \sigma x z(t)\u27e9$ remains steady at $h=1.94$ mm, but significantly fluctuates between a dominating low-stress state and a series of high-stress peaks under both $h=1.10$ and $0.31$ mm. The peak stresses of $\u27e8 \sigma x z(t)\u27e9$ at $h=0.31$ mm are close to 200 Pa, approximately twice those at $h=1.10$ mm, suggesting that the size-dependent flow curves were associated with enhanced flow heterogeneity.

To address the difference between $ t c 1$ and $ t c 2$, we investigated the propagation of local high-stress domains. Figure 4 shows the snapshots of $ \sigma xz$ evolving during representative high-stress events for the two gap sizes. At $h=1.10$ mm, a millimeter-sized high-stress front emerges from the left and then moves rightward to cross the imaging field [Fig. 4(a) and Video 1 in the supplementary material], aligning with the shear direction. In contrast, we observed reversed propagations of high-stress regions at $h=0.31$ mm, with these regions initially appearing from the right and then traveling to the left, as shown in Fig. 4(b) for two typical events (Video S2 in the supplementary material). This counter-shear propagation with a traveling period ( $ t c 2\u2248 T plane/10$) was common for $h<0.5$ mm. Compared with previous observations in cornstarch-water suspensions [28,37], the counter-flow of high-stress regions was more stable in confined PS-water suspensions.

As illustrated in Fig. 5, we propose two flow mechanisms to interpret the propagations of high-stress domains in different directions. The forward propagation mode can be understood through a similar mechanism to that proposed in [39], which explains the flow heterogeneity in dense cornstarch-water suspensions. Specifically, for the PS-water suspensions at $h=1.10$ mm, we conjecture that a locally jammed, solidlike particle aggregation moves along the shear direction [Fig. 5(a)]. Under no-slip boundary conditions, the propagation speed of this particle aggregation remains close to that of the shear plate. Consequently, a high-stress domain emerges once every rotational period ( $ t c 1\u2248 T plane$) [39]. In contrast, the counter-flow propagation of high-stress regions has been observed less frequently. This flow behavior was previously reported in transient states of cornstarch-water suspensions [28,37], possibly due to the accumulation of particles near the bottom plate [37]. In our experiments, we speculate that the counter-flow propagation at $h=0.31$ mm is associated with a simultaneous accumulation-release process of particle aggregations. As illustrated by Fig. 5(b), the particle aggregations can be partially jammed by confinement at $h=0.31$ mm. While the particles at the leading front migrate forward and gradually separate from the aggregation [yellow arrows in Fig. 5(b)], the trailing particles are stopped by this locally jammed region [red arrows in Fig. 5(b)], which results in a propagation of high-stress regions against the shear direction. As this backward propagation of boundary stresses is determined by the particle density, the propagation speed can be significantly higher than that of the shear plate ( $ t c 2\u2248 T plane/10$). In both scenarios [Figs. 5(a) and 5(b)], we expect enhanced local dilations of the high-stress domains [28,39].

### C. Glass-oil suspensions

We further performed BSM measurements on glass-oil suspensions. Since the rheological behaviors of glass-oil suspensions were insensitive to the boundary roughness [Sec. III and Fig. S5(b) in the supplementary material], the nonmonotonic flow curves remained unchanged in the BSM setup [Fig. 6(a)]. The material properties of glass-oil suspensions have two advantages for BSM characterization: (1) the nonvolatile solvent allows *in situ* measurements over a long period, and (2) the glass-oil suspensions exhibit a broader shear-thickening range than the PS-water suspensions. Thus, for glass-oil suspensions, we were able to conduct BSM measurements by systematically varying the shear stress within shear thickening regime.

Figure 6(a) shows that the nonmonotonic flows of glass-oil suspensions emerge as the gap size decreases from $h=1.68$ to $0.87$ mm in the BSM setup. We maintained the global shear stress constant at $ \tau a=1$, 2, 3, 4, and 6 Pa, respectively. Figure 6(b) presents the temporal evolutions of the averaged local stress $\u27e8 \sigma x z\u27e9$ under the confined condition, $h=0.87$ mm. Similar to PS-water suspensions, the local stress $\u27e8 \sigma x z\u27e9$ in glass-oil suspensions also fluctuates between a nearly zero-stress state and a series of high-stress peaks within the nonmonotonic regime. As $ \tau a$ increases from 1 to 4 Pa, the high-stress peaks gradually rise from 15 to 30 Pa. Representative stress maps are shown in Fig. 6(c). However, as $ \tau a$ further increases to 6 Pa, the flow heterogeneity notably reduces and $ \sigma x z$ remains constant around 7.8 Pa (Video S4 in the supplementary material). We interpret this homogenization of shear flows at $ \tau a=6$ Pa as a dynamic yielding process in that the high-stress domains relax under shear. Consequently, this process substantially reduces flow resistance [Fig. 6(a)] and decreases normal stresses sharply [Fig. 1(e)].

At a large gap size $h=1.68$ mm, the local stress fluctuations remain insignificant throughout the shear thickening regime, and the BSM measurements show a nearly homogenous shear flow (Fig. S7 in the supplementary material). These findings further confirm that the flow heterogeneity depicted in Fig. 6 was induced by boundary confinement.

Using the autocorrelation analysis employed in Eq. (9), we quantified the characteristic timescales between two consecutive high-stress peaks for confined glass-oil suspensions ( $h=0.87$ mm). Figure 7(a) represents the results of $ R \sigma $ evaluated at $ \tau a=1$, 3, and 4 Pa. When $ \tau a=1$ Pa, the first peak in $ R \sigma (\delta t)$ is located at $\delta t=192$ s, which well approximates the time interval between two adjacent peaks ( $ t c 1 \u2032$) in Fig. 7(b). For $ \tau a=3$ and 4 Pa, $\u27e8 \sigma x z(t)\u27e9$ oscillates more rapidly with two distinct timescales. For example, we observed a local maximum at $\delta t=62$ s followed by a major peak at $\delta t=104$ s. As shown in Fig. 7(b), the longer timescale corresponds to the intervals between two major peaks ( $ t c 1 \u2032\u2248106$ s), whereas the shorter timescale approximates a sub-interval between a lower spike and a major peak ( $ t c 2 \u2032\u224862$ s). The two intervals are indicated by the black and pink arrows.

To interpret the timescales $ t c 1 \u2032$ and $ t c 2 \u2032$, we also measured the dynamics of the high-stress domains in the confined glass-oil suspensions. Figure 8 shows the propagation of representative high-stress domains for $ \tau a=1$ and 4 Pa, respectively. These local domains always travel in the same direction as the shear plate, and no counter-flow propagation was observed in any experiments (Videos S5 in the supplementary material). We conjecture that the simultaneous accumulation and release of high-stress domains [as illustrated in Fig. 5(b)] are absent in glass-oil suspensions due to the relatively weak stress localizations ( $\u27e8 \sigma x z\u27e9<30$ Pa) in the nonmonotonic regime.

We herein interpret the dependence of $ t c 1 \u2032$ and $ t c 2 \u2032$ on $ T plane$ [Eq. (11)] based on the flow mechanisms shown in Fig. 5(a). When $ \tau a=1$ or 2 Pa, briefly exceeding the onset of shear thickening, a local particle aggregation forms in the suspensions. Due to the weak boundary stresses ( $\u27e8 \sigma x z\u27e9<30$ Pa), the particle aggregation possibly travels along the center of the shear cell without sticking to the shear plate [24]. The mean traveling speed of the aggregation is approximately half that of the shear plate, such that the high-stress region finishes one complete round during two rotational periods, $ t c 1 \u2032\u22482 T plane$. For $ \tau a=3$ or 4 Pa, a second high-stress domain can appear in a different location within the suspension. We, thus, observed the high-stress regions twice as frequently as we did at lower stresses, leading to $ t c 2 \u2032\u2248 T plane$.

## V. CONCLUSIONS

By systematically characterizing the shear thickening behaviors of both nonsedimenting PS-water suspensions and sedimenting glass-oil suspensions, we have demonstrated the critical role of boundary confinement in determining shear thickening rheology. The major findings of this work are summarized as follows.

Although nonmonotonic flow curves have been predicted theoretically, they have rarely been reported in previous experiments due to flow instabilities [33,40]. Our results show that strong confinement imposed by a shear boundary can effectively induce stable $S$-shaped responses in the shear thickening regime (Fig. 1). Although high-stress domains develop in the nonmonotonic regimes, boundary confinement prevents the stress heterogeneity from evolving into large-scale rheochaos [28].

Using BSM (Fig. 2), we characterized the rich dynamics of local stress heterogeneity induced by boundary confinement. For both PS-water and glass-oil suspensions, we observed the propagations of high-stress domains at speeds controlled by the rotational shear plate [Eqs. (10) and (11)]. For PS-water suspensions, we observed a crossover from the shear-directed to counter-shear propagations of high-stress domains as the gap size reduced (Fig. 4). While both propagation directions have previously been reported in dense suspensions [24,28,36,37,41], we demonstrated that these local high-stress domains remain stable under confinement. Intriguingly, the counter-shear propagation in PS-water suspensions appears to be significantly faster than the rotation of the shear plate [Eq. (10)], potentially caused by a simultaneous accumulation-release process of high-stress aggregations [Fig. 5(b)]. For glass-oil suspensions, however, the high-stress domains always propagate along the shear direction, regardless of the gap size (Fig. 8). The absence of counter-shear propagation is possibly attributed to the weak boundary stresses ( $ \sigma x z\u223c30$ Pa) induced in confined glass-oil suspensions.

The rheology of PS-water suspensions depends on the material properties of shear boundaries. With shear plates made of roughened glass, $S$-shaped flow curves appeared for confined PS-water suspensions [Fig. 1(b)]. However, when a gel film was coated on the bottom plate for BSM measurements, the flow curves were consistently monotonic even under strong confinement [Fig. 3(a)]. We attribute this absence of nonmonotonic responses to the smoothness of gel surfaces and the deformations of gel films under large boundary stresses ( $ \sigma x z\u223c300$ Pa) in confined PS-water suspensions.

In summary, we presented an experimental investigation that revealed the underlying connection between shear thickening rheology and local flow structures in confined dense suspensions. We have demonstrated that boundary confinements can effectively stabilize $S$-shaped flow curves and induce local stress heterogeneity. Our results provide valuable insights into controlling suspension rheology through boundary effects.

## SUPPLEMENTARY MATERIAL

See the supplementary material for the following: (1) the mathematical details of the BSM measurements and (2) supplementary figures with supporting data.

## ACKNOWLEDGMENTS

We thank Professor Ryohei Seto for valuable suggestions. This work was supported by the Early Research Scheme (No. 26309620), General Research Funds (Nos. 16307422 and 16305821), and the Collaborative Research Fund (No. CY6004-22Y) from the Hong Kong Research Grants Council. The research activities were also funded by the Partnership Seed Fund (No. ASPIRE2021#1) from the Asian Science and Technology Pioneering Institutes of Research and Education League and Hong Kong-Macau-Guangdong Industrialization Fund from Guangdong Science and Technology Department (No. 2023A0505030017).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.