Abstract
Colloidal dispersion has elastic properties due to Brownian relaxation process. However, experimental evidence for the elastic properties, characterized with normal stress differences, is elusive in shearing colloidal dispersion, particularly at low Péclet numbers (Pe < 1). Here, we report that single micrometer-sized polystyrene (PS) beads, suspended in silica nanoparticle dispersion (8 nm radius; 22%, v/v), laterally migrate and form a tightly focused stream by the normal stress differences, generated in pressure-driven microtube flow at low Pe. The nanoparticle dispersion was expected to behave as a Newtonian fluid because of its ultrashort relaxation time (2 μs), but large shear strain experienced by the PS beads causes the notable non-Newtonian behavior. We demonstrate that the unique rheological properties of the nanoparticle dispersion generate the secondary flow in perpendicular to mainstream in a noncircular conduit, and the elastic properties of blood plasma–constituting protein solutions are elucidated by the colloidal dynamics of protein molecules.
INTRODUCTION
Colloidal dispersion, a mixture of small particles subject to Brownian motion and their suspending liquid, is ubiquitously found in natural and biological phenomena and also in a wide range of industrially relevant products, such as food, paints, cosmetics, and electronic materials (1). Understanding the flow behaviors or rheology of such colloidal dispersions as nanoparticle-laden suspension is a prerequisite to successful material formulation and processing design (1). For instance, the rate-dependent shear viscosity, such as shear-thinning or shear-thickening behavior, substantially affects the transportation of the colloidal dispersion through pipes; the maximum flow rate is limited by the occurrence of the shear thickening (1–5). On the other hand, the colloidal dispersion tends to restore a well-dispersed equilibrium state because of the relaxation process driven by Brownian motion and interparticle forces when the spatial distribution of colloidal particles is dehomogenized by flows, which is the source of its viscoelasticity (1). Many extravagant non-Newtonian flow phenomena such as rod climbing, usually found in viscoelastic polymeric materials, can be attributed to the normal stress differences generated in shearing flow (6). However, despite vast rheological tests, experimental evidence of such fluid-mechanical behaviors induced by the normal stress differences is still elusive for colloidal dispersions composed of rigid spherical particles, particularly at Brownian motion–dominant low–Péclet number (Pe) conditions (5).
The lack of particle orientability as well as deformability, in the colloidal dispersion of rigid spheres, leads to normal stress differences too weak to experimentally detect (5, 7). However, the presence of the normal stress differences has been long predicted, even at both dilute particle volume fractions and low Pe limits (2, 3). The Péclet number represents the relative ratio of the hydrodynamic shear to thermal forces in a shear flow:
Here, we report the presence of unique non-Newtonian phenomena in the pressure-driven microtube flow of nanoparticle colloidal dispersion, which apparently shows Newtonian rheological properties. First, we found that noncolloidal particles are focused along the channel centerline. The distinctive non-Newtonian behavior, of particle focusing in nanoparticle dispersion, is analyzed and found to be caused by the normal stress differences. Furthermore, we scrutinized the lateral particle migration to examine the predictions by the existing colloidal theories: N1 and N2 have positive and negative signs in a shearing colloidal dispersion at low Pe (2, 3), respectively. In addition, the relaxation time of the nanoparticle dispersion, one of the essential material properties to characterize its elasticity, was measured by analyzing the particle focusing process, using the theoretical prediction based on a second-order fluid model (6). Further, we observed the generation of secondary flow in perpendicular to mainstream, eventually resulting in a single line focusing in a straight square channel, which suggests that N2 plays a substantial role in transporting particles in the nanoparticle dispersion. Last, the non-Newtonian elastic property of blood-constituting protein solutions was analyzed with the lateral particle migration in a microfluidic device to determine the origin of blood plasma viscoelasticity (10), as recently reported. The analyses demonstrated that the colloidal dynamics in the plasma-constituting major protein solutions lead to their elastic properties.
RESULTS
Particle focusing in nanoparticle colloidal dispersion in a circular microtube
We observed the lateral motion of single noncolloidal polystyrene [PS; radius (aps), 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 (φp), 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 qpeak = 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 (Rec = 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 Rec 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 Rec (5). Therefore, the particle focusing along the channel centerline cannot occur in Newtonian fluid irrespective of Rec. 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
(A) Schematic diagram for the noncolloidal particle PS bead (6 μm diameter; 0.01 volume %) focusing experiments in a nanoparticle dispersion (nominal volume fraction, 22 volume %; 16.2 nm diameter; LUDOX HS-40) in a circular tube (inner diameter, 25 μm). (B) SAXS analysis of the nanoparticle dispersion (LUDOX HS-40). (C) Top: Development of particle focusing of the PS beads in a capillary tube at a flow rate of 20 μl hour−1 (Pec = 0.002). The probability distribution functions (PDFs) were obtained on the basis of the centers of particles, and the labels denote the distance from the inlet. Bottom: Distribution of particles according to flow rates (50 to 200 μl hour−1 denoted as labels; 0.005 ≤ Pec ≤ 0.02) at a location 8 cm downstream from the inlet. The images were acquired by the z projection of 2000 images using the min intensity mode in ImageJ software (see Materials and Methods for the details of imaging). a.u., arbitrary units.
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 ≤ Pec ≤ 0.02), implying that the spatial distributions of colloidal particles are close to their thermal equilibrium state, where the channel Péclet number Pec is defined by
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 ul (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)
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 (φp) (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 zp along the streamwise direction and then recorded the outermost radial location (rp) of the PS beads at each zp. The initial position, ri, close to the wall is expected to gradually approach the centerline, and the evolution equation for the PS bead trajectory (rp, zp) can be derived by integrating Eq. 1 as follows (19)
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
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 (Fe) exerted on a spherical particle, which is semi-empirically modeled as Fe~ −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 (aps = 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
(A) Schematic diagram for the secondary flow–assisted single-line focusing in a square channel and an overlay of the postulated streamlines for the secondary flow generated by the second normal stress difference (N2). (B) Distributions of PS beads in a square straight PDMS channel (w × h, 25 μm × 25 μm; length, 5 cm) for the flow rates ranging from 20 to 100 μl hour−1 at a location 4.8 cm downstream from the inlet. The images were acquired by the z projection of 2000 images using the min intensity mode in ImageJ software (see Materials and Methods for the details of imaging). (C and D) Visualization experiments to investigate the mixing occurrence in Newtonian fluid (C) (65 wt % glycerin aqueous solution; viscosity, 15.2 mPa·s) and nanoparticle dispersion (D) (viscosity, 14.5 mPa·s; LUDOX HS-40) in a square channel. The same three fluid streams flowed into a square straight PDMS channel (w × h, 25 μm × 25 μm; length, 4 cm) except for the presence of the fluorescent dye; the fluorescent dye was added to the two side streams and not to the central stream (total flow rate in the straight region: 50 μl hour−1). The fluorescence intensity profiles were measured along the cross-stream direction at locations 0.2, 2, and 3.8 cm downstream from the junction point of the three streams.
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 aeff/a increases (2, 27). The effective colloidal radius aeff is introduced to accounts for the interparticle repulsion so that the distance between particles is maintained to be >2aeff (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 (φp = 0.21;
(A) SAXS analyses of DI water and PBS-added nanoparticle dispersions (final PBS concentration ≈ 1× PBS;
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).
(A) Schematic diagram for the PDMS microchannel used for the lateral particle migration experiments in protein solutions. The data and error bars denote average values and standard deviations, respectively (n = 5). (B) PDFs in 4% (w/w) BSA solution in DI water, Newtonian fluid [15% (w/w) glycerin solution in DI water], and 4% (w/w) BSA solution in 1× PBS solution at a flow rate of 10 μl hour−1. (C) PDFs in 1.5% (w/w) γ-globulin (γ-G) solutions and in a mixture of 1.5% (w/w) γ-globulin and 4% (w/w) BSA in 1× PBS at a flow rate of 10 μl hour−1. The data and error bars denote average values and standard deviations, respectively (n = 3).
Here, PS beads (aps = 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 (Rg) of 5.3 nm (36), and the radii of IgM and IgA are larger than or comparable to that of IgG [also compare Rg = 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.
DISCUSSION
This study demonstrated particle (noncolloidal sized) focusing in pressure-driven microscale flow in a nanoparticle dispersion. This phenomenon was elucidated by considering the particle migration induced by the gradients of the normal stress differences. We report that colloidal dynamics, in the model nanoparticle dispersion, drives the very weak elastic property that is represented by an ultrashort relaxation time [O(1) μs]. The current results suggest the possible occurrence of size-based particle segregation because of the normal stress differences induced by the colloidal dynamics during material processing, when the material is composed of multisized particles from nano- to micrometer, e.g., lithium-ion battery electrode materials. In addition, we expect that the single-line focusing formation, regardless of the channel cross-sectional shape (Figs. 1 and 2), will find practical use in lab-on-a-chip applications such as particle counting and sorting (23). Last, we mention that the blood plasma–constituting proteins are much more diverse than those in the current studies, and the protein species or the protein-protein interaction that have not been investigated in this study may generate stronger viscoelasticity than the current cases. Nonetheless, understanding that blood plasma viscoelasticity can be expressed by the colloidal dynamics of blood plasma–constituting major proteins is an important result. We expect that the studies on the relationship between the composition of the blood plasma–constituting proteins and the viscoelastic properties will open novel routes to understanding the blood cell dynamics, e.g., the lateral cell distribution affected by the elastic property originating from the protein colloidal dynamics that may be affected by human health conditions.
MATERIALS AND METHODS
Materials
A commercial nanoparticle dispersion (LUDOX HS-40; Sigma-Aldrich), composed of electrostatic repulsion–stabilized silica nanoparticles (40%, w/w) dispersed in aqueous solution, was used as received, in which the particle volume fraction (
Microfluidic setups and imaging
Particle migration experiments were performed in a cylindrical fused silica microtube [inner radius (R), 12.5 μm; length, 10 cm; PEEKSil Tubing, catalog no. 62510, Upchurch] or PDMS channels. PDMS channels were fabricated as in our previous study (39). For the particle migration experiments, we used almost the same microfluidic setup as in our previous study (19) except for slight modifications: The flow rate was controlled with a syringe pump (11 Plus, Harvard Apparatus), and the images were captured with a high-speed camera (MC2, Photron) installed on an inverted optical microscope (IX71; Olympus) equipped with ×20 objective (×1.6 internal magnification was applied), and the acquired images were analyzed with ImageJ software (National Institutes of Health). All the particle locations in both fused silica tube and PDMS channels were analyzed as in the previous study (19) except for the 6-μm PS beads in the fused silica tube, which were determined by manually checking the center of each bead with the ImageJ software, because of the low optical contrast ratio of the PS beads near the channel wall. The time-lapse images were stacked in z projection with the “min intensity” or “standard deviation” option in the ImageJ software to qualitatively demonstrate the extent of the particle focusing; the min intensity option was selected to show maximally scattered particle locations near the channel centerline, whereas the standard deviation option was chosen to qualitatively present the average particle distribution (19). The contrast and brightness for the dark image, stacked with the min intensity option in ImageJ software, was evenly enhanced. For the mixing visualization experiments in a square PDMS channel, the same three fluid streams were flowed into a square PDMS channel (w × h, 25 μm × 25 μm; length, 4 cm) except for the presence of the fluorescent dye: A small amount of fluorescent dye (250 ppm) was added to the two side streams and not to the central stream. The two side streams emanated from an inlet, and the total flow rate of the two side streams and the central stream flow rate were equally controlled with the two equivalent syringes (500 μl, Hamilton) equipped on a syringe pump (11 Plus, Harvard Apparatus). The images were acquired with a charge-coupled device camera (DMK 23U445) installed on an inverted optical microscope (IX71, Olympus) equipped with ×20 objective. We used the same fluorescence experimental setups and data analysis procedure as in the previous study (40).
SUPPLEMENTARY MATERIALS
Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/6/eaav4819/DC1
Supplementary Text S1. Particle pressure–driven lateral migration.
Supplementary Text S2. Measurement of relaxation time using particle migration.
Supplementary Text S3. Inertial particle migration.
Fig. S1. Distributions of PS beads according to flow rates, particle sizes, and colloidal particle volume fractions.
Fig. S2. Measurement of the relaxation time of nanoparticle dispersion (LUDOX HS-40).
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REFERENCES AND NOTES
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