Normal stress difference–driven particle focusing in nanoparticle colloidal dispersion

Particle focusing in nanoparticle colloidal dispersion in a circular microtube

We observed the lateral motion of single noncolloidal polystyrene [PS; radius (a_ps), 3 μm; volume fraction, 0.01% (v/v)] beads suspended in a commercially available nanoparticle dispersion (LUDOX HS-40) flowing through a capillary tube [inner radius (R), 12.5 μm], as schematically shown in Fig. 1A. The nanoparticle dispersion (used as received) is composed of electrostatic repulsion–stabilized silica nanoparticles [mean radius (a), 8.1 nm; coefficient of variance (CV), 0.14; nominal volume fraction of nanoparticles (φ_p), 0.22; refer to Materials and Methods for the detailed characterization procedure of the nanoparticle dispersion], dispersed in aqueous solution. We confirmed the well-dispersed state of the nanoparticle dispersion with a peak at q^peak = 0.31 nm⁻¹ in the small-angle x-ray scattering (SAXS) data, corresponding to an average distance of ≅ 20 nm due to the repulsive interaction between the silica nanoparticles (Fig. 1B). Previous rheological tests demonstrated that the nanoparticle dispersion (LUDOX HS-40) behaves as a Newtonian fluid due to its constant shear viscosity and nondetectable linear viscoelastic properties (11). However, from the experiments (Fig. 1C; see also fig. S1), it is readily seen that the dynamics of PS beads in the nanoparticle dispersion are substantially different from those in Newtonian fluid, because the PS beads are tightly focused along the channel centerline. PS beads are gradually focused at a flow rate (Q) of 20 μl hour⁻¹, while the beads move from an inlet to downstream (Fig. 1C, top). The channel Reynolds number (Re_c = l〈u〉ρ/μ) was very small (≃ 0.03), where l, 〈u〉, ρ, and μ are the characteristic length scale (l = 2R for the current circular tube), average velocity in the cross section in a channel, density, and viscosity of the nanoparticle dispersion, respectively. On the contrary, no lateral particle motion is expected in a Newtonian fluid at zero Re_c because of the so-called time-reversibility condition (5) or the equilibrium particle position is determined at the middle between channel centerline (≈0.6R) by shear gradient and wall lift forces at finite Re_c (5). Therefore, the particle focusing along the channel centerline cannot occur in Newtonian fluid irrespective of Re_c. In addition, the particle focusing observed in the nanoparticle dispersion cannot be attributed to shear-induced migration, which may arise from the interaction among the PS beads (12), because of the very small volume fraction of the PS beads (0.01%, v/v). The measured shear viscosity of the nanoparticle dispersion was constant as 14.5 mPa·s for shear rates up to γ̇ = 1000 s⁻¹ at 20°C, when measured with a rotational rheometer equipped with cone-and-plate geometry (1°, 60 mm diameter), and the Newtonian behavior in shear viscosity is consistent with a previous study (11). We performed additional viscosity measurements using parallel-plate geometry with small gap heights (40 mm diameter; gap heights, 20, 30, 50, and 100 μm) to enhance the maximum accessible shear rate range (13), which also shows Newtonian behavior up to γ̇ = 20,000 s⁻¹ (Pe ≅ 0.05) (the relative viscosity difference was less than 5% when measured with the parallel-plate geometry having a gap height of 20 μm, as the shear rate was changed from 100 to 20,000 s⁻¹).

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PS beads form a tightly focused stream up to higher flow rates (200 μl hour⁻¹), and the particle focusing is slightly intensified as the flow rate increases (Fig. 1C, bottom). All the experiments presented in Fig. 1C were performed under low–Péclet number conditions (0.002 ≤ Pe_c ≤ 0.02), implying that the spatial distributions of colloidal particles are close to their thermal equilibrium state, where the channel Péclet number Pe_c is defined by Pec=6πμsγ̇ca3/kBT and γ̇c is the characteristic shear rate 〈u〉/R. The particle focusing found in the nanoparticle dispersion manifests the non-Newtonian behavior of colloidal dispersion under low–Péclet number conditions not found in colloidal rheology studies (1–5).

We elucidate that the normal stress differences (N₁ and N₂) cause the particle migration and focusing in the nanoparticle dispersion. Particle migration, driven by the normal stress differences, was originally proposed to explain the lateral particle motion observed in the pressure-driven flow of viscoelastic polymer solutions (8, 9). Ho and Leal (8) predicted with a second-order fluid model that the lateral migration of a rigid spherical particle can be induced by the combined gradients of N₁ and N₂. In a circular tube, the lateral migration velocity u_l (r/R) of a spherical particle, subject to the pressure-driven flow in a second-order fluid, was predicted as the following equation (14–16)ul(r/R)〈u〉=−αWi(1−βN2N1)(apsR)2(rR)(1)where Wi is the Weissenberg number (≡λγ̇c; liquid-like when Wi ≪ 1, solid-like when Wi ≫ 1) and λ is the relaxation time of the suspending fluid (6). Brunn (15) predicted that α and β have positive values of 1.84 and 2.83, respectively. In the equation, the normal stress differences, N₁ and N₂, were modeled as N1/μsγ̇=0.6(1−φp/φm)−3Pe and N2/μsγ̇=−0.42(1−φp/φm)−3Pe (φm; maximum packing fraction), respectively, following previous theoretical models in the low Pe limit for concentrated hard-sphere colloidal dispersions (3, 17) (hence, N₂/N₁ is assumed to be −0.42/0.6 in Eq. 1), which may demand further improvement for the repulsive nanoparticle dispersion. However, experimental validation of the predicted N₁ and N₂ is still challenging, especially at low Pe, because of their small magnitudes (7). Furthermore, it is not even clear that N₁ and N₂ have positive and negative signs, respectively, as predicted (2, 3, 17). However, the positiveness of N₁ was observed just before the occurrence of the shear thickening, where Pe was non-negligibly large (Pe ≥ 65) (18). In Eq. 1, the combined effects of positive N₁ and negative N₂ predict the inward particle migration at low Pe, akin to the conventional particle migration in viscoelastic polymer solutions with constant shear viscosity (19, 20). Together, the predictions, based on existing colloidal theories (Eq. 1), are consistent with our experimental observations (Fig. 1), supporting the predicted positive and negative signs for N₁ and N₂ in colloidal dispersion at low Pe. However, we note that “particle pressure” (5) may play an important role in moving the PS beads suspended in the nanoparticle dispersion at high Pe, which was not considered at the current low Pe conditions (refer to text S1).

Additional experimental data demonstrated that particle dynamics in the nanoparticle dispersion can be successfully explained with Eq. 1. The particle focusing is intensified with increasing PS bead size or flow rate for a fixed-tube radius, but it is attenuated with decreasing colloidal volume fraction (φ_p) (refer to fig. S1). Meanwhile, the particle focusing process in the nanoparticle dispersion can be harnessed to characterize its relaxation time similarly as in previous studies on polymer solutions (refer to text S2 and fig. S2 for the detailed procedure and its validation). We observed the radial distribution of the PS beads at each observation location z_p along the streamwise direction and then recorded the outermost radial location (r_p) of the PS beads at each z_p. The initial position, r_i, close to the wall is expected to gradually approach the centerline, and the evolution equation for the PS bead trajectory (r_p, z_p) can be derived by integrating Eq. 1 as follows (19)2lnrpri+(riR)2−(rpR)2=−αWi(1−βN2N1)(apsR)2(ΔzR)(2)where Δz is defined as z_p − z_i, and it was assumed that the streamwise velocity of a PS bead is equal to the fully developed Newtonian velocity profile at the PS bead location (19, 20). The outermost location of PS beads (a_ps = 1.2 μm; a_ps/R ≅ 0.1) was defined similarly as in our previous work (19): the radial location, within which 99% of the particles are distributed near the channel centerline, is defined as r_p at the axial location z_p. The left-hand side in Eq. 2, termed the “focusing index” (19), and the relaxation time of the nanoparticle dispersion were calculated to be 2.0 ± 0.1 μs (mean ± SD; n = 3) by fitting the focusing index versus Δz/R with Eq. 2 for the given flow and geometrical conditions, where z_i is located at 1 cm downstream from the inlet. On the other hand, the relaxation time of colloidal dispersion is defined by the characteristic time required to relax the disturbed colloidal particle distribution to its equilibrium state (1, 21). It was demonstrated that the mean longest relaxation time of colloidal dispersion, which was obtained through the linear viscoelastic properties measured in small amplitude shear flow, is comparable to the characteristic Péclet time scale [τp≡a2/6D0s(φp)] for diffusion over a particle radius (a) (21), where D0s(φp) is the short-time self-diffusion coefficient at colloidal particle volume fraction φp(D0s(φp)→D0 as φ_p → 0). D₀ is the free-solution self-diffusion coefficient given by the Stokes-Einstein relation (D0≡kBT/6πμsγ̇a). In our experimental condition (φ_p = 0.22), τp is predicted to be 0.73 μs according to the theoretical prediction of the short-time diffusion coefficient [D0s(φp)/D0=1−1.832φp−0.219φp2] for hard-sphere dispersion (22). Our measured relaxation time of 2 μs, determined by analyzing the particle focusing process, does not notably deviate from the estimated diffusion time scale τp = 0.73 μs, which suggests that the main relaxation process in the nanoparticle dispersion originates from the Brownian motion of colloidal particles, as previously observed (21).

Until now, the fluid-mechanical phenomena related to the non-Newtonian elastic property at low Pe were usually considered to be hard to detect, which can be attributed to the equivalently low-Wi condition at low Pe. For instance, even the maximum Wi is estimated to be only O(10⁻³) in the present experimental conditions due to its ultrashort relaxation time of 2 μs. Therefore, this nanoparticle dispersion can be regarded as having a very small degree of elasticity and behaving similarly to Newtonian fluid. Therefore, the distinctive non-Newtonian phenomenon of the observed particle focusing is apparently counterintuitive, but it is thought to be possible because the trace particles (PS beads) undergo a very large shear strain as estimated to be ≃∫γ̇dt≈γ̇cttravel≈(〈u〉/R)(L/〈u〉)=6000 while they travel through the microchannel, where t_travel denotes the traveling time of a PS bead through the channel and L corresponds to the channel length.

Particle focusing in a square channel: Secondary flow effect

We reported that particle focusing in the nanoparticle dispersion is driven by the combined effects of the first and second normal stress differences (N₁ and N₂), where the magnitudes of the two normal stress differences are quite comparable (17). This suggests that N₂ plays an important role in migrating the PS beads because −βN₂/N₁ = 1.98 (15, 17) > 1 in Eqs. 1 and 2. On the contrary, N₂ is usually neglected in polymer solutions due to its much smaller magnitude compared with N₁ (20). We further investigated the N₂ effects on the particle focusing and the flow dynamics in a square microchannel, which has non-axisymmetric flow field in the channel cross section. It was demonstrated in polymer solutions that multiple equilibrium particle positions along the centerline and four corners are formed, where the shear rates along these five locations are all zero (consequently N₁ = 0) (23). This is because the particles are driven toward these positions by the elastic force (F_e) exerted on a spherical particle, which is semi-empirically modeled as F_e~ −a³ ∇ N₁ (20). On the other hand, it was numerically predicted that the multiple equilibrium positions in a square channel can be reduced to single-line focusing by the combined effects of the normal stress difference–driven particle migration and the cross-stream secondary flow generated by N₂ (24). Therefore, we postulated that the nontrivial N₂ in the nanoparticle dispersion may change the multiple equilibrium particle positions to a single particle stream within the cross section of a square channel, as schematically drawn in Fig. 2A. We investigated the spatial distribution of the 0.01% (v/v) PS beads (a_ps = 3 μm) flowing through a straight, square, poly(dimethylsiloxane) (PDMS) channel [width × height (w × h), 25 μm × 25 μm; length, 5 cm]. As shown in Fig. 2B, single-line focusing is formed along the channel centerline in the nanoparticle dispersion (LUDOX HS-40), which was observed for a range of flow rates (20 to 100 μl hour⁻¹) (0.0008 ≤Pec≡6πμsγ̇ca3/kBT≤ 0.004; 0.02 ≤ Re_c ≡ l〈u〉ρ/μ ≤ 0.1), where the characteristic length (l) and shear rate (γ̇c) are defined by l = 2h and γ̇c≡2〈u〉/h. The single-line focusing in the square channel supports the generation of the secondary flow. In addition, to more directly validate the secondary flow generation in the nanoparticle dispersion, we investigated dye mixing in a square microchannel (w × h, 25 μm × 25 μm; length, 4 cm), as shown in Fig. 2 (C and D). The same three fluid streams except for the presence of the fluorescent dye flowed into the square channel: A small amount of fluorescent dye [250 parts per million (ppm)] was added to the two side streams and not to the central stream. We observed that the fluorescence intensity profile in the nanoparticle dispersion gradually became homogenized as the fluid streams moved downstream (Fig. 2D), which contrasts with the constant profiles irrespective of the axial location in a Newtonian fluid (Fig. 2C), corroborating the generation of a cross-stream secondary flow. N₂ is considered to cause the secondary flow (25, 26) in noncircular tubes, which was also recently found in non-Brownian suspensions (26). The secondary flow generation, found in the nanoparticle dispersion, suggests that N₂ plays an important role in the flow dynamics of the nanoparticle dispersion and, consequently, affects the transportation of the suspended particles.

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Particle migration in blood plasma–constituting protein solutions

Particle migration and focusing in the model colloidal dispersion (LUDOX HS-40) suggests that non-Newtonian fluid properties may appear in biological colloidal dispersions, such as protein solutions, through the colloidal dynamics. However, electrolytes are abundant in biological fluids, which may substantially change the colloidal dynamics by screening the electrostatic repulsion interaction between colloidal particles (1). Colloidal dispersions such as the nanoparticle dispersion, in which the van der Waals attractive force is relevant, are practically stabilized with electrostatic repulsion and/or steric hindrance, as in the current nanoparticle dispersion (1). In a repulsive colloidal dispersion, the contribution of interparticle repulsive forces to the total normal stress differences may dominate over Brownian and hydrodynamic forces between colloidal particles as the ratio of a_eff/a increases (2, 27). The effective colloidal radius a_eff is introduced to accounts for the interparticle repulsion so that the distance between particles is maintained to be >2a_eff (2). To understand the effects of the electrostatic repulsion screening on the elastic properties, we investigated particle migration in the nanoparticle dispersion mixed with an electrolyte solution [20× phosphate-buffered saline (PBS) = 19:1; LUDOX HS-40], of which the ionic strength is assumed to be comparable with 1× PBS (ionic strength, 163 mM). As a reference, a slightly diluted dispersion with deionized (DI) water (LUDOX HS-40/DI water, 19:1) was also prepared to match the volume fraction (φ_p = 0.21; φeff≡(aeff/a)3φp = 0.36), where φeff denotes an effective volume fraction (see Materials and Methods for how to determine φeff). All the experiments in the PBS solution were performed within 1 hour after PBS was added to the original nanoparticle dispersion, because the aggregation of colloidal particles progresses slowly (28). The PBS addition substantially reduced the repulsive interactions, as shown in the SAXS and shear viscosity data (Fig. 3, A and B), and consequently, φeff was reduced to 0.25, which is quite close to its nominal volume fraction, φ_p = 0.21. In the PBS-added solution, we found no discernible particle focusing at a flow rate of either 50 or 100 μl hour⁻¹ (Fig. 3, C and D), indicating the weakening of the elastic property by the screening of electrostatic repulsion under the physiological conditions (2, 27).

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Fundamental studies on particle focusing in the model colloidal dispersions provide a clue for understanding the origin for the elastic properties of human blood plasma (10), as recently observed, which has not yet been well understood mainly due to its very short relaxation time (10⁻³ to 10⁻⁵ s) (29). The elastic nature of human blood plasma was identified only in the extensional flows of the capillary breakup extensional rheometer and contraction-expansion geometry, in which the stretching dynamics of the polymer may play a key role in generating viscoelasticity (10, 29). However, the shear flow characteristics are dominant over the extensional flow in blood vessel flow. In addition, the extent of polymer stretching in shear flow is limited because of its rotational component compared to that in the extensional flow (30). Therefore, the behaviors of protein molecules as rigid colloidal particles would be more pronounced than the behaviors of flexible polymers in shear flow. Blood plasma is a complicated aquatic mixture composed of diverse inorganic and organic materials including many different protein families such as albumin, globulin, and fibrinogen, where the globulin is again subcategorized into four subclasses: α 1, α 2, β, and γ types (31). Here, we focus on two major proteins (albumin and γ-globulin) in blood plasma, where γ-globulin is the richest globulin type (32). First, we examined the lateral particle migration in a bovine serum albumin (BSA) solution in the microfluidic device presented in Fig. 4A. Albumin is the most abundant among the proteins that constitute blood plasma. Early studies demonstrated that the BSA solution behaved like a yield stress fluid, the origin of which was attributed to the lattice structure formed by repulsive interactions among BSA molecules, as found in SAXS experiments (33, 34). However, a more recent study has suspected that the highly non-Newtonian characteristics of the BSA solution, observed in rotational rheometer experiments, may originate from the interfacial viscoelasticity caused by the accumulated proteins at the liquid-air interface (35).

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Here, PS beads (a_ps = 3 μm) were transported through the microfluidic channel depicted in Fig. 4A in DI water and 1× PBS solution at 4% (w/w) BSA concentration, a concentration that is within physiological conditions [reference range, 3.5 to 5.5 g dl⁻¹ (32)]. We note that there is no liquid-air interface as the solutions are being injected into the microchannel from a syringe. We observed that the PS beads in DI water were noticeably aligned along the channel centerline, while no notable particle focusing was detected in the BSA solution in PBS solution (Fig. 4B). For reference, we also demonstrated the lateral particle distribution in a Newtonian fluid [15% (w/w) glycerin solution in DI water], which also showed no noticeable particle focusing along the channel centerline (Fig. 4B) (refer to text S3). The experimental results indicate the notable elasticity of the BSA solution in DI water, which is substantially reduced in PBS, akin to the experimental observation in the model nanoparticle dispersion. The non-elastic property observed in the PBS solution implies that BSA is not solely responsible for the blood plasma elasticity in the physiological condition, which is seemingly the same conclusion as the previous study (10), in which BSA aqueous solution was examined for the blood plasma elasticity (10).

Next, we investigated the particle focusing in bovine γ-globulin [reference range, 0.7 to 1.7 g dl⁻¹ (32)] solutions in PBS solution. The protein γ-globulin is not just one kind of protein but rather is a mixture of immunoglobulins: IgG (80%), IgM (10%), and IgA (<10%) (36). The size of the most abundant subclass, IgG, was found to have a radius of gyration (R_g) of 5.3 nm (36), and the radii of IgM and IgA are larger than or comparable to that of IgG [also compare R_g = 2.8 nm for BSA (37)]. As shown in Fig. 4C, the particle focusing is pronounced in 1.5% (w/w) γ-globulin solution in PBS solution. Therefore, experimental results suggest that γ-globulin may play an important role in generating the elasticity of blood plasma. It was recently observed that dimers formed by short-range interactions are abundant in γ-globulin solution (36), and monoclonal IgG1 formed “large, loosely bound, transient clusters” in electrolyte solutions (38) different from BSA. From the viewpoint of colloidal dynamics, we propose that the elastic properties of the γ-globulin solution originate from (i) the large sizes of γ-globulin molecules or dimers [N₁, N₂ ~ a³ (3, 17)], as compared with BSA (36, 37), and (ii) the dynamic network structure formed by the interaction among γ-globulins (38). The particle distributions in the γ-globulin solution have shoulders (Fig. 4C), which may be attributed to the formation of dynamic clusters possibly subject to the local shear rate; this demands further study. The particle focusing was further intensified when 4% (w/w) BSA was added to a 1.5% (w/w) γ-globulin solution in PBS (see Fig. 4C). The enhanced elasticity observed in the γ-globulin and BSA mixture suggests the formation of complexes between γ-globulin and BSA molecules, which implies that the elastic properties of blood plasma are affected by complicated protein interactions among different proteins.