Passive non-linear microrheology for determining extensional viscosity

Bulk measurements of stress and viscosity are essential to understand the non-equilibrium behavior of complex fluids. Macroscopic stress-strain relations in structured materials such as polymer solutions or colloidal suspensions are known to arise due to molecular-scale behavior and dynamic microstructural effects. To this end, flow-induced particle migration is a phenomenon that illustrates the connection between bulk stresses and microscopic interactions in fluids. Particle migration has long been observed in various complex fluids including suspensions,^1–3 polymer solutions,^4–6 and multi-component materials such as blood flows that exhibit margination,^7,8 known as the Fahraeus-Lindqvist effect.⁹ 

Flow-induced migration can arise due to several different phenomena, including hydrodynamic interactions with boundaries,¹⁰ inertial effects,¹¹ or viscoelastic effects due to stress gradients in flowing fluids.^12,13 Particle migration in viscoelastic materials was first experimentally observed in shear flow by Karnis and Mason,⁴ where it was found that spherical particles suspended in polyisobutylene (PIB) solutions migrated towards the direction of minimum shear rate, which corresponds to the channel center-line in a pressure-driven Poiseuille flow. Leal and co-workers used a second-order fluid model to quantitatively understand this phenomenon and attributed migration to normal stress due to lateral variations in shear rate.^12,13 In addition to polymer solutions, particle migration has also been studied in colloidal suspensions.^14–17 Morris and co-workers developed an experimental apparatus to measure the osmotic pressure change resulting from shear-induced particle migration in particulate suspensions,¹⁸ where it was found that changes in pressure varied linearly with shear rate and were related to normal stress. Particle migration in polymer solutions has also been studied using microfluidics,¹⁹ including methods to determine relaxation times in polymer solutions in uniform channels.²⁰ Recently, a new microfluidic method to precisely focus particles was demonstrated using a combination of inertial and elastic focusing effects.²¹ Despite recent progress on studying particle migration phenomena in flow, however, the vast majority of prior work on flow-induced migration has nearly exclusively focused on shear flow or uniform flow in channels.

Unlike shear flow, purely extensional flows consist only of extensional/compressional character with no elements of fluid rotation, which is known to generate strong stress responses in polymeric materials. The flow properties of polymer solutions in extensional flow have been studied using rheo-optical techniques such as flow birefringence,^22–28 filament stretching rheometry,^29–31 capillary breakup extensional rheometry (CABER),^32–37 drop break-up in microchannels,^38–41 jetting-based extensional rheometry,^42,43 and dripping-onto-substrate (DoS) extensional rheometry.^44–46 Ober et al. developed a microfluidic-based “indexing” method to estimate extensional viscosity by measuring mechanical pressure drop across a hyperbolic contraction geometry using on-chip fabricated pressure sensors;⁴⁷ however, given the complex mixture of shear and extensional flow components in this geometry, it is challenging to determine extensional viscosity using the approach.^47,48 Pressure drop measurements across cross-slot flow geometries are generally more amenable to determining extensional viscosity by separating the effects of shear flow and extensional flow,^43,49 though recent work has highlighted difficulties in separating the contributions of shear and extension-generated normal stresses in several microfluidic geometries.⁴⁸ Taken together, there are ongoing challenges in using microfluidics to measure extensional flow properties for complex fluids.

In this work, we directly observe particle migration in semi-dilute polymer solutions in planar extensional flow (Fig. 1). Microfluidic experiments are performed in the regime of negligible inertia (Reynolds number $Re≪1$⁠), such that particle migration occurs due to finite normal stresses in the flowing polymer solution. Particle migration trajectories are analyzed in the context of a second-order fluid model,^12,13 which enables determination of steady extensional viscosity under moderate flow rates. In our work, planar extensional flow is generated in a cross-slot microfluidic device using pressure-driven flow, resulting in a two-dimensional velocity profile in the x-y plane with a strain rate $𝜖 ̇ ( z )$ that varies in the transverse direction (z-direction) (Fig. 1). Single tracer particles suspended in polymer solutions are trapped in two-dimensions near the stagnation point using an active feedback control method known as a Stokes trap.^50,51 In semi-dilute polymer solutions, particles are observed to migrate towards the channel walls in the direction transverse to flow (z-direction), which is consistent with prior experimental⁴ and theoretical studies^12,13 of particle migration in shear flow showing that particles tend to migrate in the direction of decreasing shear rate due to normal stress effects. Using this approach, we present the direct observation of particle migration in viscoelastic polymer solutions in extensional flow, coupled with detailed analysis of these experimental data in order to determine extensional viscosity using microfluidics.

We prepared three different semi-dilute polymer solutions based on a biopolymer (DNA) and a synthetic polymer (polyethylene oxide, PEO). Semi-dilute DNA solutions (λ-DNA, Invitrogen, 48.5 kbp, $M=3.2×1 0 7$ g/mol) are prepared in viscous aqueous solutions (60% w/w sucrose, 30 mM Tris-HCl, pH 8.0, 0.1 mM EDTA, and 5 mM NaCl) at two different concentrations corresponding to 1 c^* and 2.5 c^*, where $c *$ is the polymer overlap concentration (⁠$c *$ = 40 μg/ml for λ-DNA⁵²), as previously described.⁵³ The solvent viscosity η_s of the 1 c^* and 2.5 c^* DNA solutions was 55 cP and 60 cP, respectively (Table I). Semi-dilute solutions of PEO (Sigma-Aldrich, average $M w =1.0×1 0 6$ g/mol) are prepared in aqueous solutions at a concentration of 10 c^* (1.7% w/w). In all cases, polymer solutions are well-mixed and homogeneous, and experiments are conducted using a circulating water bath thermally coupled to microfluidic devices to maintain a constant temperature (⁠$22 °$C).

Prior to microfluidic experiments, the bulk rheological properties of DNA and PEO solutions were determined, including zero-shear viscosity η₀ (Table I) and steady shear viscosity (Fig. S1 of the supplementary material). Zero-shear viscosity measurements and linear viscoelastic measurements were performed using a Discovery Hybrid Rheometer 3 (40 mm parallel plate geometry). Flow sweep zero-shear viscosity measurements on 1 c^* λ-DNA, 2.5 c^* λ-DNA, and 10 c^* PEO solution were conducted, and zero-shear viscosity η₀ is measured and reported as shown in Fig. S1. Linear viscoelastic measurements (frequency sweep measurements) on 10 c^* PEO solutions are shown in Fig. S2 of the supplementary material. For semi-dilute DNA solutions, longest relaxation times τ were determined using single molecule analysis by direct observation of single polymer relaxation,⁵³ together with the scaling relation $τ/τo∼(c/c*)0.48$⁠, where τ_o is the longest polymer relaxation time in dilute solution. For semi-dilute PEO solutions, the longest relaxation time is obtained by linear viscoelastic measurements and DoS rheometry.

Two-layer polydimethylsiloxane (PDMS) microfluidic devices are fabricated using standard techniques in soft lithography, such that a fluidic layer is situated below a control layer that contains a pressure-controlled valve.⁵³ The fluidic layer is designed to contain a cross-slot flow geometry that generates a planar extensional flow (Fig. 1). In this flow field, the velocity $v$ in the x-y plane is given by $[ v x , v y , v z ] = [ − 𝜖 ̇ ( z ) ( x − x o ) , 𝜖 ̇ ( z ) ( y − y o ) , 0 ]$⁠, where (x_o,y_o) is the location of the stagnation point and $𝜖 ̇ ( z )$ is the strain rate. Fluidic microchannels have a width of 300 μm and a depth of 100 μm in the cross-slot region. Single particles are trapped in flow near the stagnation point using a feedback controlled technique known as a Stokes trap.⁵¹ The details of the implementation of the trapping method have been presented in prior work.^50,53 In brief, single particles are trapped in a 2D extensional flow using the following procedure: (1) images of fluorescent particles in flow are captured using a charge-coupled device (CCD) camera, and particle positions are detected using a custom LabView program for image analysis, (2) a target particle (e.g., fluorescent particle detected closest to the stagnation point) is selected for trapping, and (3) the target particle is confined in the flow plane (x-y plane) by actuating the on-chip membrane valve situated above one outlet channel (Fig. 1), which enables fine-scale modulation of the location of the stagnation point via slight changes to the fluidic resistance in one outlet flow line with respect to the other in the 4-channel cross-slot geometry. In this work, the action of the membrane valve results in negligible changes in the strain rate $𝜖 ̇$⁠.⁵³ 

Before studying particle migration in extensional flow, particle tracking velocimetry (PTV) was used to determine flow field kinematics and strain rates $𝜖 ̇$ in polymer solutions (Fig. 2), which allows for determination of a dimensionless flow strength known as the Weissenberg number $Wi= 𝜖 ̇ τ$⁠. A trace amount of fluorescent microbeads (0.84 μm diameter, Spherotech, 0.01% v/v) was added to polymer solutions to enable particle tracking. Microfluidic devices are mounted on an inverted fluorescence microscope (Olympus IX71), which allows for real-time imaging of fluorescent beads using a high numerical aperture (1.45 NA, 100$×$⁠) oil-immersion objective lens and a solid-state CW laser (Coherent, 488 nm). Polymer solutions are introduced into microfluidic devices using a pressure regulator (Proportion Air), and images of bead positions are acquired as functions of applied pressure and z-position using a CCD camera (AVT Stingray). A custom particle tracking and analysis program is used to determine strain rates $𝜖 ̇$ for all polymer solutions, which allows for determination of a dimensionless flow strength known as the Weissenberg number $Wi= 𝜖 ̇ τ$⁠. Unless otherwise stated, Wi values are defined based on the strain rate $𝜖 ̇$ at the channel mid-plane. In all cases, the flow field was observed to be stable with no evidence of elastic instabilities.⁵⁴ Moreover, the strain rate profile in the z-direction was found to be parabolic (Fig. 2), which suggests small departures from Newtonian flow behavior at low to moderate Wi.

Following flow field characterization, we studied particle migration in extensional flows of semi-dilute polymer solutions. The experiment begins by introducing a polymer solution containing a trace amount of fluorescent particles (0.84 μm diameter) into a microfluidic device via pressure-driven flow. Single particles are trapped in flow using a Stokes trap, and particle migration in the transverse direction (z-direction) is tracked using a piezo-nanopositioning stage (Physik Instrumente) and a custom control scheme (Fig. 3). In brief, the particle tracking algorithm is used to control the piezo stage to maintain a trapped fluorescent particle in focus during migration in the z-direction. The piezo stage is first initialized near the center plane of the microdevice (within 1–2 μm), and the area of the trapped particle is measured via an image analysis routine in LabView. As the particle migrates in the z-direction and begins to drift out of the focal plane, the piezo stage adjusts its position (up or down) such that the detected area of the particle is minimized. In this way, the z-position of the particle is recorded during the experiment. For reference, we show a few characteristic single particle migration trajectories from our experiments, together with the measured particle area as a function of time (Fig. S3 of the supplementary material). In these experiments, the particle position is sampled at 10 Hz.

Dripping-onto-substrate (DoS) experiments are used to determine the bulk extensional viscosity of polymer solutions, as previously reported (Fig. 4).^44,45 The necks formed between a nozzle and a substrate for the PEO and DNA solutions are cylindrical slender filaments, characteristic of high viscosity or high elasticity fluids,^34,44,45,55 which are distinct from conical necks associated with inviscid fluids or power law fluids.^45,56 The values of stress σ/R(t) and extension rates $𝜖̇=−2(dlnR(t)/dt)$ are calculated from radius R(t) evolution data, where representative plots are shown in Figs. S4–S6 of the supplementary material. Transient and steady extensional viscosities are obtained from the radius evolution data using the following equation:

Extensional viscosity obtained using capillary-thinning based methods such as CaBER,^32,34 DoS rheometry,^44,45 and jetting-based rheometry⁴³ are typically plotted as a function of the Hencky strain or the total accumulated strain $𝜖=2ln R o / R ( t )$ in the liquid filament that continuously increases as the neck radius R(t) thins over time. Radius evolution data in the final regime before pinch-off for the aqueous PEO solutions and DNA solutions show a linear decrease with time, which can be captured by visco-capillary (VC) scaling relation,^57–59 

where R_o is the initial radius, t_p is the pinch-off time, and η_E is the extensional viscosity. The presence of the terminal regime allows for the measurement of terminal steady extensional viscosity that is independent of both extension rate and strain. Theterminal steady extensional viscosity value is arguably a true material response^60,61 that depends on both molecular weight and polymer concentration, but not on the extension rate or the accumulated strain.^61–63 

In order to quantitatively understand microfluidic experiments, we sought to develop an analytical model to describe the migration of particles in viscoelastic solutions in an extensional flow. The migration of rigid spheres in a general quadratic flow of a second-order fluid was previously considered by Leal and Chan using analytical arguments.¹³ Their results show that particle migration occurs due to normal stresses whenever a lateral variation in shear rate occurs in the undisturbed flow.¹² The second-order fluid model is useful for slowly varying flows of viscoelastic fluids (⁠$Wi≲$ 1), and it provides a rational basis for separating the effects induced by normal stresses from those due to viscosity-dependent flow rates, mainly because the shear dependence of viscosity only enters at third-order in the expansion.⁶⁴ For these reasons, we adapted the theory of Leal and Chan¹³ to the case of particle migration in planar extensional flow with a z-dependent strain rate.

An analytical expression for the dimensionless migration velocity $( Ũ s ) i$ of a rigid sphere in a general quadratic flow of a second-order fluid is given by¹³ 

where $𝜖 i j k$ is the Levi-Civita tensor and e_ij, ψ_ijk, and $θ i j$ are the symmetric and irreducible tensorial components of the velocity coefficient tensors β_ij and γ_ijk for a general quadratic flow. Here, an irreducible tensor is defined to contract to zero along any two indices, such that β_ii = γ_iji = γ_iij = 0.

To obtain the particle migration velocity $( Ũ s ) i$⁠, the individual terms in Eq. (3) need to be determined. To proceed, we introduce an expression for a dimensionless general quadratic velocity in a fixed frame,

where $α i ′$⁠, $β i j ′$⁠, and $γ i j k ′$ are the constant velocity coefficient tensors and $x ̃ i ′$ is the dimensionless position vector relative the fixed frame (Fig. 5). For this derivation, it is preferred to adopt a coordinate system with the origin at the center of the particle, such that the coordinate system translates but does not rotate with the flow.¹³ The dimensionless position vector in the translating coordinate frame is $x ̃ i$⁠. In this way, $x ̃ i ′ = x ̃ i + ( x ̃ 0 ( t ) ) i$⁠, where $( Ũ s ) i = ( x ̃ 0 ( t ) ) i$ is the position vector for the center of rotation of the particle measured in the fixed frame, as shown in Fig. 5. Substituting this relation into Eq. (4), we obtain

where we have used the symmetry condition $γ i j k ′ = γ i k j ′$⁠. We can now define a new set of velocity coefficient tensors in the translating frame α_i, β_ij, and γ_ijk,

where α_i and β_ij are time-dependent due to the translating coordinate frame, but γ_ijk is constant in time. We now arrive at an expression for the velocity in the fixed frame,

or, alternatively, an expression for the velocity in the translating frame,

Our goal is to solve for β_ij and γ_ijk in Eq. (8). The coefficients are related to the velocity field $V ̃ i$ in the following relations:

Following Chan and Leal,¹³ β_ij and γ_ijk can be decomposed into their respective irreducible components as follows:

Using this approach, we applied the analytical theory for particle migration in a second-order fluid to planar extensional flow. To begin, we define the locations of the fixed frame coordinate system given by (x′,y′,z′) and the translating coordinate system given by (x,y,z), as shown in Fig. 5. The origin of fixed frame is located at the center of the cross-slot in the x-y plane and at the bottom of the microdevice, such that z′ = 0 corresponds to the bottom wall. The particle is located an arbitrary distance δ away from the bottom wall. Clearly, the fixed frame and translating frame are coincident in the x-y plane, but not in the z-direction. The z-components of the coordinate systems are related by the relation: $z ′ =δ+l z ̃$⁠, where $z ̃$ is the dimensionless coordinate in the z-direction in the translating frame, rendered dimensionless with the microchannel height l. The expression for the dimensional undisturbed velocity field in the fixed frame is $[ v x ′ , v y ′ , v z ′ ] = [ − 𝜖 ̇ ( z ′ ) ( x ′ − x o ′ ) , 𝜖 ̇ ( z ′ ) ( y ′ − y o ′ ) , 0 ]$⁠, where $( x o ′ , y o ′ )$ is the location of the stagnation point and the z-dependent strain rate is $𝜖 ̇ ( z ′ ) =4 𝜖 ̇ m a x ( z ′ / l ) ( 1 − z ′ / l )$⁠, shown schematically in Fig. 5 and validated by particle tracking experiments as shown in Fig. 2. The dimensionless strain rate can be expressed in the translating frame as $𝜖 ̇ ̃ ( z ̃ ) =4 𝜖 ̇ ̃ m a x ( α + z ̃ ) ( 1 − α − z ̃ )$⁠, where α = δ/l.

The problem reduces to determining the tensors e_ij, ψ_ijk, and $θ i j$ for the undisturbed velocity profile. To proceed, we solve for these terms using a dimensionless velocity profile $[ v ̃ x , v ̃ y , v ̃ z ]$ and the dimensionless strain rate profile $𝜖 ̇ ̃ ( z ̃ )$⁠. The individual terms in the particle migration velocity given by Eqs. (3) and (10) are shown in Table II. Here, we define i = 1 to be the x-axis, i = 2 to be the y-axis, and i = 3 to be the z-axis. Combining the above relevant terms in Table II, including e_mnψ_mni, $𝜖 i m n 𝜖 m l θ l n$⁠, and e_imτ_m, and Eq. (3), we obtain an expression for the particle migration velocity in the z-direction in a planar extensional flow,

To simplify the analysis, we define the quantities $N 1 ( 𝜖 1 ) = 10 27 ( 5 + 13 𝜖 1 ) + 4 27 ( 1 + 11 𝜖 1 )$ and $N 2 ( 𝜖 1 ) = 2 27 ( 5 + 13 𝜖 1 ) + 1 54 ( 1 + 11 𝜖 1 ) + 1 6$⁠. Moreover, $Δ x ̃ = ( x ̃ − x ̃ 0 )$ and $Δ y ̃ = ( y ̃ − y ̃ 0 )$ are assumed to be a trapped particle’s instantaneous distance from the stagnation point $x ̃ 0$ and $y ̃ 0$ in the 2D plane. We assume that trapped particles exhibit symmetric and minor excursions from the stagnation point in the x-y plane, such that $Δ x ̃ =Δ y ̃$ = 50% of particle diameter a resulting in $2Δ x ̃ 2 =0.5$⁠, which is consistent with prior experiments using the automated trap.⁶⁵ In this way, $( x ̃ − x ̃ 0 ) 2 + ( y ̃ − y ̃ 0 ) 2 =2Δ x ̃ 2$⁠, and Eq. (11) can be expressed as