Polymers in Microflow
CHAPTER 4 POLYMERS IN MICROFLOW
4.4 Cross-Streamline Migration
4.4.1 Inertial Migration
In vertical tubes, heavier particles migrate in upwards flow toward the walls, whereas lighter particles migrate toward the centerline [104, 105]. For neutrally buoyant rigid spheres, experiments by Segré and Silberberg showed an accumulation at a radial position of approximately 0.6 pipe radii of the circular tubes (diameter: ≈ 1cm) [99].
These experiments were performed in the laminar flow regimes with Reynolds numbers Rec based on the channel width d between 2 and 700. The results are confirmed by other experimental studies, for all of which the Reynolds number was of order unity or higher [104, 106, 107]. For example, Goldsmith and Mason showed that rigid particles are homogenously distributed across the channel for very low Reynolds numbers, and they migrate to a position between the wall and the centerline for finite Reynolds numbers
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[108]. While the former studies were all performed at the macroscopic scale with large tube dimensions (typically millimeter-centimeter in diameter), recent experiments showed that inertial migration can also be significant in microfluidics [100]. In microchannels (width d ≈ 50µm), 9µm large spheres migrate to an equilibrium position in flow with velocities of about m/s. In this case, the particle Reynolds number is of order of one [100]. The particle Reynolds number in Poiseuille flow based on the size of the particle (diameter a of the sphere) and the averaged shear rate is given by
d a Rep vo
η ρ 2
= , 4.43
where v0 is the maximal velocity of the fluid at the centerline.
Figure 4-6: Migration of a sphere in simple shear flow at finite Reynolds numbers. The velocity of the sphere vs differs from the velocity of the fluid. The dashed green arrows illustrate the fluid flow relative to the sphere. a) A sphere moving slower than the fluid experiences a force in the direction of larger flow velocities. b) A sphere moving faster than the fluid experiences a force in the direction of lower flow velocities.
The experimental results for all studies mentioned above have been explained by inertial effects due to finite Reynolds numbers [109-116]. Brownian motion of the particles is not included in the considerations. In 1965, Saffman [115] illustrated how fluid inertia can induce a lateral force in simple laminar shear flow for a rigid sphere having a non-zero velocity relative to the fluid. He considered the scenario of a sphere in an unbounded fluid and calculated, via matched asymptotic expansion method, the lateral
CHAPTER 4 POLYMERS IN MICROFLOW
force. This force arises from the interaction of the disturbance flow created by the sphere and the velocity gradient. The magnitude of this force is given by [115]
v Ka
Fs = 2γ&1/2(ηρ)1/2 , 4.44
where K ≈ 6.46 is a numerical constant and v is the velocity of the sphere relative to the fluid. The direction of this lateral force is always toward regions where the fluid velocity relative to the particle is larger (see figure 4-6) [103, 110]. Due to the finite size of the particle, solvent molecules have to be displaced laterally when the sphere moves with a relative velocity to the fluid in flow direction. This displacement becomes irreversible at large distances away from the sphere because of inertia. The difference in the velocity of the displaced fluid from the background flow is larger for higher relative fluid velocities. Therefore, a pressure gradient along the sphere is created which generates a force in the direction of the larger relative velocity. This means that a particle moving faster than the fluid migrates toward slower flow (see figure 4-6b).
Conversely, a particle moving slower experiences a force in the direction of the larger flow (see figure 4-6a).
Figure 4-7: Cross-streamline migration due to rotation of a sphere in flow.
In simple shear flow, a neutrally buoyant sphere moves with the same velocity as the fluid and no Saffman force is induced. However, the Saffman force is relevant for non-neutrally buoyant spheres. A heavier sphere moving upwards in a vertical tube experiences a force toward the wall, while a lighter sphere experiences a force toward the centerline. This is qualitatively consistent with the experiments mentioned above [104, 105, 117].
Additionally, rotation of a sphere can also lead to migration analogous to the Magnus effect (figure 4-7). A pressure reduction is created by the rotational velocity at the side of the sphere where the velocity increases the fluid velocity. Consequently, the sphere will migrate in this direction. This lift force is given by [118]
v a
FRB =π 3ρΩ , 4.45
where Ω is the angular velocity and v is the fluid velocity. In Poiseuille flow, the
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migration due to rotation of the sphere is always in direction toward the centerline [103].
Thus, this force cannot explain the migration of neutrally buoyant spheres away from the channel center, as was observed by Segré and Silberberg. An additional aspect has to be considered in Poiseuille flow, namely the curvature of the velocity field [109, 111, 119]. Dividing the sphere with finite size in two halves, the side toward the wall has an overall larger relative velocity compared to the side toward the centerline (see figure 4-8). Consequently, the sphere migrates toward the channel wall by using the same argument considered for the Saffman force. If the sphere comes close to the wall, it is repelled from the wall due to inertial interactions with the wall [110, 119, 120]. The interplay between the wall repulsion and the migration away from the centerline due to the curvature of the velocity field creates an equilibrium position at a position between the wall and the centerline. For non-neutrally buoyant spheres, an additional Saffman contribution has to be considered.
Figure 4-8: Inertial migration of a sphere in Poiseuille flow due to the curvature of the flow field.
The sphere migrates in direction toward the channel wall. The green arrows illustrate the relative fluid velocity.
Most theoretical studies are based on solutions of the Navier-Stokes equation by using perpetuation methods [109-116]. At small distances around the sphere, the disturbance flow created by the sphere is to leading order determined by the Stokes equation. The viscous forces are dominant compared to inertial forces in this regime, the so-called inner region. However, they are on the same order at large distances away from the sphere (outer region). The Saffman force (equation 4.44) has been calculated for an
CHAPTER 4 POLYMERS IN MICROFLOW
unbounded fluid for which the inertial effects in the far field are responsible for its existence. However, the walls confine the fluid inside tubes. If the walls of the tube lie in the outer region, the particle motion is influenced by the inertial effects in the far field [109, 110]. If the channel width is small enough that the walls are found in the inner region, irreversible inertial interactions with the walls are relevant. The length scales lv =ηρ/v0 =d/Rec and ls =
(
ηρd/v0)
1/2 =d/Re1c/2 determine at which distance away from the particle the inertial effects from the far field become important for Poiseuille flow [109, 110, 116]. If the channel Reynolds number is of order of one, both length scales are comparable with the channel width and inertial effects from the far field are significant. Therefore, the theoretical studies can be classified by the channel Reynolds number [110] in regimes Rec << 1 [111, 112], Rec = O(1) [109, 110, 113, 116], and Rec >> 1 [109, 114, 115]. In all of these studies, the particle Reynolds number and the ratio of sphere diameter and channel width have been assumed to be small (Rep << 1 and a/d << 1).Although the experiments showing the migration of the sphere away from the centerline are performed with channel Reynolds numbers of unity or larger, the first theoretical theories in confined geometries considered the regime Rec << 1 [111, 112]. Regular perpetuation techniques are used to obtain an expression for wall-induced inertial effects. The migration velocity can be calculated from [112]
( )
y aRe f( )
y positive (which means migration toward the wall) between the centerline and ym, and( )
yfv is negative (migration toward the centerline) between the wall and ym. Consequently, it can be expected that the particles migrate to the middle between the wall and the centerline.
For the regime Rec = O(1) in which the experiments have been performed, singular perpetuation methods are used in order to include inertial effects in the far field. The lateral force for a neutrally buoyant sphere is given by [109]
(
Re y)
Re f(
Re y)
where fA
(
Rec,y)
is a coefficient which dependent on channel position and channel Reynolds number but not on the particle size. fA(
Rec,y)
is positive at the channel center (migration toward the wall) and negative at the channel wall (migration toward the centerline). fA(
Rec,y)
decreases monotonically from the channel center toward theCHAPTER 4
wall and is zero at the equilibrium position. For rotational spheres, a correction can be made to include the migration toward the centerline due to the rotation of the sphere [109]. This correction is only small because the force due to rotation is one order of magnitude smaller than the force in equation 4.47 [110]. Balancing the lateral force with the Stokes drag forceFdrag =3πηav, the migration velocity is given by [100]
(
Re y)
aRe f(
Re y)
This expression converges asymptotically for decreasing channel Reynolds numbers toward equation 4.46 [29]. The migration velocity depends on particle size, and decreases strongly for smaller particles. This might explain why migration was only observed in the above mentioned microfluidic experiments when the particle Reynolds number is of order of one [100].
The expressions 4.46 and 4.48 have been calculated for rigid spheres of diameter a. In chapter 5, we consider the cross-streamline migration of semiflexible actin filaments (diameter ≈ 7nm, length ≈ 8µm) in aqueous solution (ρ ≈1kg/L, η ≈ 1mPa⋅s) inside microchannels (width d ≈ 10µm, length ≈ 2cm) at flow velocities of about mm/s. The Reynolds numbers based on the channel width are Rec ≈ 10-3…10-2. Considering an actin monomer (length ≈ diameter ≈ 7nm) as a rigid sphere, the migration velocity vmig ≈ 3·10-15m/s can be estimated by equation 4.46. The diameter of a sphere having the same volume V ≈ 3·10-22m3 (mass: ≈ 3·10-19kg) as a filament is d = 80nm. Such a sphere migrates with velocity vmig ≈ 5·10-12m/s. Therefore, inertial effects can be neglected for cross-streamline migration. However, a description of polymer migration has to include conformational changes due to the flow field, Brownian motion, and “particle-particle”
interactions (i.e. interactions between the segments of a single polymer).