THE JOURNAL OF CHEMICAL PHYSICS 140, 094903 (2014)
Flow-induced demixing of polymer-colloid mixtures
in microfluidic channels
Arash Nikoubashman, Nathan A. Mahynski, Amir H. Pirayandeh,
and Athanassios Z. Panagiotopoulosa)
Department of Chemical and Biological Engineering, Princeton University, Princeton,
New Jersey 08544, USA
(Received 5 December 2013; accepted 6 February 2014; published online 6 March 2014)
We employ extensive computer simulations to study the flow behavior of spherical, nanoscale
colloids in a viscoelastic solvent under Poiseuille flow. The systems are confined in a slit-like
microfluidic channel, and viscoelasticity is introduced explicitly through the inclusion of polymer
chains on the same length scale as the dispersed solute particles. We systematically study the effects
of flow strength and polymer concentration, and identify a regime in which the colloids migrate
to the centerline of the microchannel, expelling the polymer chains to the sides. This behavior
was recently identified in experiments, but a detailed understanding of the underlying physics
was lacking. To this end, we provide a detailed analysis of this phenomenon and discuss ways
to maximize its effectiveness. The focusing mechanism can be exploited to separate and capture
particles at the sub-micrometer scale using simple microfluidic devices, which is a crucial task for
many biomedical applications, such as cell counting and genomic mapping. © 2014 AIP Publishing
LLC. [http://dx.doi.org/10.1063/1.4866762]
I. INTRODUCTION
Colloidal dispersions and polymers in solution are very
interesting systems from both a thermodynamic and rheological point of view. In contrast to simple liquids, these
fluids respond to external stresses, e.g., shear or pressure gradients, in a highly nonlinear fashion, which gives rise to phenomena including shear-thinning,1, 2 shear-thickening,1 and
viscoelasticity.3 The nonlinear response to stress has its origins in the disparate length and time scales between the solvent and suspended particles. Although these non-Newtonian
liquids are ubiquitous in nature and technology, e.g., blood4
and crude oil,5 the detailed, microscopic mechanisms responsible for their nonlinear behavior still remain incompletely understood. This is especially true for multicomponent systems
such as colloid-polymer mixtures, where in addition to the
solute-solvent interactions, the complex interactions between
the different species play an important role.
To better understand these effects, the dynamics of dilute
colloid and polymer solutions have been studied extensively
in the last decade with the advent of nano- and microfluidic devices6–8 and sophisticated computer simulations.9, 10
Early studies focused on infinitely dilute suspensions of rigid
colloids and flexible polymers in a Newtonian solvent such
as toluene or hexane, which is ideal for studying the (flow)
properties of single solute particles.11–17 Among the major discoveries was the distinct cross-stream particle migration and resulting highly non-uniform solute distribution for
both rigid11, 12, 16 and soft14, 15 particles when subjected to
Poiseuille flow. This effect stems from the inhomogeneous
a) Author to whom correspondence should be addressed. Electronic mail:
[email protected]
0021-9606/2014/140(9)/094903/8/$30.00
solvent velocity, which induces a gradient in the stress field
across the channel.18 This flow-induced migration has important consequences for many applications where the goal is to
precisely control the lateral position of the solute, such as cell
counting,19 filtration,20, 21 and genomic mapping.22
In principle, it is possible to overcome this lateral drifting and to spatially focus the solute particles on a streamline
by imposing external forces, such as electric fields,23 or by
introducing obstacles.24 In practice however, these so-called
active methods often require significant modifications of the
solute or the channel, and therefore add an unnecessary layer
of complexity to the system. Alternatively, one can manipulate the lateral position of the solute particles by inducing nonlinear flows. Such conditions can, for instance, be achieved by
tuning the solvent properties. Indeed, early experiments on
microscopic particles dispersed in flowing viscoelastic media
revealed a particle migration toward the region of minimal
shear.25–31 Such a passive approach is a very auspicious candidate for spatial focusing, since it requires only minimal adjustments to the experimental setup. There is a tremendous
interest in the biomedical industry of exploiting this mechanism for nanoparticles, due to the possibility of isolating
and capturing specific cells and viruses.18, 32, 33 However, it is
not immediately clear whether macroscopic viscoelastic effects translate to the nanoscopic colloidal scale, as the effects of Brownian motion become increasingly significant in
the sub-micrometer regime. In addition, a continuum description of the surrounding viscoelastic medium becomes problematic at these length scales, since its constituent polymers
are comparable in size to the solute particles. Therefore, it
is more appropriate to regard these system as binary colloidpolymer mixtures instead. This in turn renders approaches
based on continuum fluid mechanics only partially usable,
140, 094903-1
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Nikoubashman et al.
J. Chem. Phys. 140, 094903 (2014)
and thus one has to resort to alternative methods for modeling
these systems, such as molecular dynamics (MD) simulations
employed in this work.
Indeed, only very recently the general feasibility
of viscoelastic focusing has been experimentally demonstrated for nanometric rigid colloids and DNA molecules.34
However, experiments only provide limited insights into the
system on a microscopic scale, and therefore the detailed
mechanisms responsible for this flow-induced focusing remain largely unclear. In order to better understand this behavior, we studied the flow behavior of rigid spherical colloids
dispersed in both Newtonian and viscoelastic solvents, using
extensive computer simulations that closely replicate experimental conditions. Our simulations successfully reproduce
the viscoelastic focusing onto a centerline seen experimentally at low Reynolds numbers.34 In addition, we found that
the focusing cannot be maintained indefinitely, but fails when
the flow rate exceeded some threshold value. We provide a
detailed microscopic explanation for the pathways involved
in these flow-induced transitions and discuss how the parameters can be optimized for viscoelastic focusing in microfluidic
devices.
The rest of this paper is organized as follows; in Sec. II,
we introduce our simulation model and give a brief overview
of the employed methodology. Then we present and discuss
our results in Sec. III. Finally, we summarize our findings and
provide a brief outlook for future problems in Sec. IV.
II. MODEL AND SIMULATION METHOD
Our model system consists of rigid spherical colloids
with diameter σ c at a concentration of c = πρc σc3 /6 dispersed in a solvent, where ρ c is the number density of colloids. The solvent in which the colloids are dispersed is
constructed on two length scales: an underlying Newtonian
solvent described through point-like particles simulated via
the Multi-Particle Collision Dynamics (MPCD) simulation
scheme,10, 35, 36 and explicitly modeled polymer chains that
provide viscoelasticity to the dispersion medium. The viscoelasticity of the solvent was adjusted through the polymer
length Np and concentration m = πρp Np σm3 /6, where ρ p is
the polymer number density and σ m the monomer diameter.
This multiscale approach is motivated by the large time- and
length-scale discrepancies between the (microscopic) solvent
and the dispersed particles, which make fully atomistic simulations prohibitively time-consuming. Figure 1 shows a simulation snapshot which highlights the essential features of our
model. In what follows, we provide a detailed description of
the employed model and simulation method.
We employed a bead-spring model for the polymers,
where each chain is comprised of Np beads with diameter
σ m = 1, which defines our unit of length. The bonds between two consecutive monomers are described by the finitely
extensible nonlinear elastic (FENE) interaction potential37
UFENE (r) =
− 12 kr02 ln 1 −
∞,
r2
r02
, r ≤ r0
r > r0
.
(1)
FIG. 1. Simulation snapshot of a colloid-polymer mixture (with concentration c = 0.025, m = 0.050) under Poiseuille flow at Re = 32. Colloids
are colored in red, while individual polymers are color-coded. The solvent
velocity profile along the channel is schematically shown on the right hand
side.
Here, r0 is the maximum bond length and k is the spring
constant, for which we have chosen the values r0 = 1.5 and
k = 12 ε/σm2 to avoid any unphysical bond crossing even at
high flow rates.
Excluded volume between the colloids and monomers
was modeled through the purely repulsive Weeks-ChandlerAnderson (WCA) potential,38 corresponding to good solvent
conditions
σ 12 σ 6 4ε rij
+ ε, r ≤ 21/6 σij
− rij
, (2)
Uij (r) =
0,
r > 21/6 σij
where r = |ri − rj | denotes the distance between particles
i and j. Furthermore, the parameter ε = 5 kB T controls the
strength of the mutual repulsion, while σ ij = (σ i + σ j )/2 is
the mean diameter.
Our simulations were conducted in a slit-like channel,
where the confining walls were separated by a distance of Lx
from each other with their normals parallel to the x-axis of the
system. Each confining surface is considered to be an immobile monolayer of interaction sites (similar to the monomeric
units), with homogeneous density ρσm3 = 1. Since such a detailed representation of the surfaces is computationally expensive and superfluous for the problem at hand, we employed an
effective interaction potential obtained by integrating over all
wall particles17, 39
√ σ 3
2
2 σij 9
π ε 15
− xij + 310 , x ≤ (2/5)1/6 σij
3
x
Uw (x) =
.
0,
x > (2/5)1/6 σij
(3)
In Eq. (3) above, x denotes the distance from either wall,
and the total external potential caused by both walls is given
by Uext (x) = Uw (x) + Uw (Lx − x). We set Lx = 64 (which
corresponds to Lx = 8 σ c for σ c = 8), and carefully checked
for finite-size and incommensurability effects by conducting
additional simulations with Lx = 96 (= 12 σc ) and Lx = 60
(= 7.5 σc ) at selected state points. Periodic boundary conditions were applied in both the y- and z-direction of the system,
where we chose Ly = 32 and Lz = 120 to avoid unphysical
self-interactions. For such a microfluidic channel, one often
refers to the principal axes also as the gradient- (x), vorticity(y), and flow-directions (z).
In our simulation scheme, the microscopic solvent particles are considered as ideal point particles with unit mass
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Nikoubashman et al.
J. Chem. Phys. 140, 094903 (2014)
m that do not interact through pairwise potentials with each
other and the solute but rather through stochastic momentum
exchanges. The MPCD algorithm consists of two alternating
steps, a streaming and a collision step. During the former, all
solvent particles propagate ballistically over
√ a period of t,
resulting in a mean free path of λ = t kB T /m. Then, the
particles are sorted into cells with edge length a = σ m , and
undergo collisions with solvent and solute particles within the
same cell. Here, the detailed microscopic solvent collisions
are mimicked through a collective rotation of the solvent velocities around a fixed angle α around a randomly chosen unit
vector.35 Due to the cell-based nature of this algorithm, the
spatial resolution of the hydrodynamic interactions is determined by the mesh size of the collision cells, a. It has been
shown in Ref. 40 that Galilean invariance is violated for λ
< a/2. Therefore, all lattice cells are shifted by a randomly
chosen vector, drawn from a cube with edge length in the
interval [−a/2, +a/2] before each collision step. While the
aforementioned rules governing the solvent dynamics are
general, the simulation of specific flow profiles requires special care and will be discussed briefly further below (see
Refs. 10, 17, and 43 for a more detailed description of the
simulation algorithm).
For the conventional MD-timesteps, we used a Verlet
integration schemewith tMD = 2 × 10−3 measured in the
unit of time τ = mσm2 /(kB T ). For the MPCD algorithm,
we chose a collision angle of α = 130◦ and a timestep of
t = 4 × 10−2 , which leads to a mean free path of λ = 0.04.
The density of the solvent was set to ρs = 10 σm−3 , resulting in
a total number of Ns ≈ 2.5 × 106 solvent particles. These parameters lead to a dynamic viscosity of η = 19.7 τ kB T /σm3 ,
which was determined by fitting the velocity profile of the
pure solvent under Poiseuille flow through Eq. (4). Furthermore, we density-matched the rigid colloids and monomers
with the surrounding solvent to achieve neutrally buoyant
conditions; each monomer has a mass equal to the average
solvent mass per MPCD cell, Mm = 10, while we assigned
Mc = ρs π σc3 /6 to each colloid. For experiments conducted
at room temperature and channel widths of a few micrometers, these parameters describe a solvent with a viscosity of
η ≈ 0.3 cP (comparable to hexane41 ). The systems were first
equilibrated for 2.5 × 106 timesteps under quiescent conditions and then for 107 timesteps with the flow turned on. We
found that this period was sufficient to reach a steady state
for all investigated systems. Then we ran the simulations for
additional 107 timesteps to perform the actual measurements.
In the case of a pure Newtonian solvent confined in such
a microfluidic channel, a parabolic velocity profile emerges
between the two confining plates when a pressure drop is applied along the z-direction due to the no-slip boundary conditions at the confining interfaces (see Fig. 1). We emulated
this behavior by applying a volume force (similar to gravity)
to the solvent particles. The functional form of the velocity
profile is given by
vz (x) =
g
(Lx − x)x,
2ν
(4)
where g is an acceleration constant that can be used to tune
the strength of the flow, and ν = η/ρ s denotes the kine-
matic solvent viscosity. The velocity profile vanishes at the
channel walls (x = 0 and x = Lx ), and attains its maximum
value, vmax , at the center of the channel (x = Lx /2). The mean
solvent velocity can be obtained by averaging Eq. (4), yielding v0 = 2vmax /3. The parabolic velocity profile further implies that a local shear rate γ̇ (x) = ∂vz (x)/∂x is established
in the fluid, expressed as
γ̇ (x) =
g
(Lx − 2x).
2ν
(5)
Finally, we can then derive from Eq. (5) the average shear
rate between the centerline and either channel wall as γ̇ = gLx /(4ν).
In MPCD, no-slip boundary conditions can be achieved
for planar walls coinciding with the collision cell boundaries
by employing a bounce-back rule for the solvent particles.
However, for a more general system geometry the walls may
not coincide with, or even be parallel to the collision cell
boundaries. Furthermore, partially occupied collision cells
can also emerge from the cell-shifting, which is unavoidable for the small mean free paths λ employed in our simulations. Lamura et al.42 have demonstrated that in such a
case the bounce-back rule has to be modified by refilling the
boundary cells with virtual particles, and we employed this
algorithm in our simulations. Additionally, thermostatting is
required in any non-equilibrium MPCD simulation whenever
either isothermal conditions are required or viscous heating
can occur. Therefore, we rescaled the solvent velocities at the
cellular level to maintain constant temperature.43 Finally, special care has to be taken when setting the flow strength to
avoid any significant compressibility effects due to the ideal
nature of the solvent particles. Hence, the maximum fluid velocity vmax should
√ be well below the speed of sound in the
medium cs = 5kB T /(3m). We found that the Mach number, Ma = vmax /cs , was less than 0.15 for the highest flow
strengths employed in our simulations. This is well below the
threshold Ma = 0.25 above which compressibility effects become significant,41 and therefore we can consider our solvent
to be an incompressible liquid.
In order to quantify the strength of the flow, we introduce
the Reynolds number Re, which is a measure of the ratio of
the inertial forces to the viscous forces, and define it as
Re =
L3
Lx v0
= g x2 .
ν
12ν
(6)
In the limit of Re 1, i.e., Stokes flow, neutrally buoyant
solute particles are confined on a streamline and do not exhibit any cross-stream migration, because of the linearity and
thus time reversibility of the system.44 However, for stronger
flows (1 ≤ Re ≤ 100) the reciprocity of the system is broken,
and lateral lift forces lead to a distinct cross-stream migration
away from the channel’s centerline.11, 12, 15, 16 The strength of
these lift forces is highly dependent on the size and the position of the solute particles in the microchannel, and expressions have been determined analytically by asymptotic
expansion for small solute particles,45 by simulations,16, 46
and microfluidic experiments.46
These inertially driven lateral forces are however
suppressed for solute particles dispersed in viscoelastic
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Nikoubashman et al.
media;25–31 these systems exhibit an inhomogeneous stress
under flow due to the polymeric nature of the solvent, which
results in a solute migration toward the region of lowest γ̇ .
For steady Poiseuille flow, this region corresponds to the centerline, where the local shear-rate vanishes completely. Expressions for the viscoelastic lift force have been derived for
dispersed particles with σ c 1 μm,28–30 but these findings
are not readily applicable to nanoparticles, since Brownian
motion plays a significant role at these length scales.30 Another complication arises from the fact that nanometric solute
particles have a comparable size as the polymers constituting
the viscoelastic solvent. Hence, it is more appropriate to regard the system as a colloid-polymer mixture than as colloids
embedded in a continuous viscoelastic medium.
An important measure for polymer solutions is the Weissenberg number Wi = τp γ̇ , which is the product of polymer
relaxation time τ p and (local) shear-rate. It has been shown
in Ref. 47 that the MPCD algorithm correctly reproduces
Zimm dynamics for sufficiently small solvent mean free paths
λ 0.1. Since we have λ = 0.04 in our simulations, we
can safely assume that our polymers conform with the Zimm
model, and therefore use the following expression for τ p :48
τp,n =
3
ηRg,0
.
(7)
kB T n3γ
Here, τ p,n is the relaxation time of the nth mode, and
γ = 0.588 is the Flory exponent for good solvent conditions.
For steady shear, the dynamics of the polymer are dictated by
its longest relaxation time, and therefore we can approximate
τ p ≈ τ p,1 .
III. RESULTS
In order to elucidate the origin and properties of
nanoscale viscoelastic focusing for colloid-polymer mixtures
in microfluidic channels under Poiseuille flow, we systematically varied the colloid and polymer concentrations (0 ≤ c
≤ 0.075 and 0 ≤ m ≤ 0.075) and the flow strength (0 ≤ Re
≤ 128). In addition, we also studied the influence of the colloid diameter (σ c = 6, 8, 10) and polymer length (Np = 5, 10,
20, and 40). If not otherwise stated, we employed σ c = 8 and
Np = 40, where the polymers had an equilibrium radius of
gyration 2Rg,0 ≈ σ c . For the highest investigated concentrations with colloids of diameter σ c = 8 and chains of length
Np = 40, these values correspond to 69 colloids and 880
polymers.
We first studied the behavior of pure systems of colloids
or polymers, i.e., m = 0 or c = 0, in order to verify our
algorithm and establish a reference for the mixed systems.
Figure 2 schematically shows the particle distributions along
the gradient direction of the flow as a function of Re and concentrations. In the pure colloid case, the solute particles were
evenly distributed in the channel at zero flow [see Fig. 2(a)].
When the flow was turned on, colloids migrated toward the
channel walls and formed hexagonally packed layers in the
flow-vorticity plane close to the surfaces due to the shearinduced inertial lift forces. As Re was increased, the colloids
were pushed further to the walls and formed a more tightly
packed hexagonal structure at the surfaces. In the case of a
J. Chem. Phys. 140, 094903 (2014)
FIG. 2. Schematic representation of the density distributions ρ(x) of colloids (continuous, red) and polymers (dashed, blue) along the gradient direction (x) of the flow. (a), (d), and (g) correspond to the pure colloid case
(m = 0), (b), (e), and (h) correspond to the composite system, and (c), (f),
and (i) correspond to the pure polymer case (c = 0). (a)–(c) show ρ(x) at
quiescent conditions, while (d)–(f) and (g)–(i) show the system at Reynolds
numbers below and above the critical value Rec , respectively.
pure polymer solution, we observed a homogeneous density
distribution ρ p (x) along the x-direction under quiescent conditions [see Fig. 2(c)]. As we slowly increased the flow strength,
polymers migrated away from the centerline to the walls as
shown in panels (f) and (i) of Fig. 2. This cross-stream migration is caused by the inhomogeneity of γ̇ (x), which leads to
a position-dependent deformation and shear-alignment of the
polymers. This in turn leads to a gradient in the chain mobility
which manifests itself in the partial evacuation of the centerline as Re is increased. These results confirm previous studies of colloidal dispersions11, 12, 16 and polymer solutions13–15
under flow in the limit of infinite dilution, and we can therefore conclude that the characteristic features at → 0 are
indeed qualitatively preserved for the significantly higher investigated here.
However, upon mixing the colloids and polymers, the resulting density profiles are not superpositions of the individual
ρ(x), indicating a strong interaction between the two species.
Indeed, it is already well established through experiments49
and theory50, 51 that, under quiescent conditions, athermal
colloid-polymer mixtures phase separate due to depletion effects at sufficiently high concentrations [see Fig. 2(b)]. This
purely entropic phenomenon arises as colloids approach one
another, excluding polymers from existing in the gap between
them, resulting in an osmotic pressure difference between the
bulk and locally depleted region which drives the colloids to
aggregate. The phase behavior of this binary mixture strongly
depends on the size ratio between the two components, generally classified in terms of q = 2Rg,0 /σ c . The so-called “colloid” limit, where q 1, has been more extensively studied
than its counterpart the “protein” limit (q 1), since the polymer’s internal degrees of freedom play a less consequential
role in the overall free energy landscape at densities where
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Nikoubashman et al.
0
0
0.25
0.5
x / Lx
0.75
1
FIG. 3. Colloid density ρ c along the gradient direction for a colloid-polymer
mixture with c = 0.025 and m = 0.050.
phase separation typically occurs.51, 52 We chose to focus our
efforts in this regime as well, since it is experimentally the
more common of the two.
As we turned on the flow, a significant fraction of colloids migrated to the centerline of the microchannel, thereby
expelling the polymers toward the confining walls. This configuration is schematically shown in Fig. 2(e), while Fig. 3
shows the density profiles from simulations. Here, we can
clearly identify a colloid- and polymer-rich region with a very
sharp interface between them. This flow-induced demixing
occurred for all investigated polymer lengths and particle concentrations, with some quantitative differences which will be
discussed further below. As we increased the flow strength
above some critical Rec , we measured a steady evacuation
of the centerline with a concomitant increase of colloids in
the vicinity of the channel walls, until the channel center was
entirely devoid of colloids [cf. Fig. 2(h)].
In order to understand the origin of the flow-induced
colloid focusing, let us consider a system where the colloids are initially placed in the centerline of the channel
[cf. Fig. 2(e)]; in the case of a Newtonian solvent, inertial
lift forces would immediately push the solute particles toward
the channel walls upon turning on the flow.11, 12, 15, 16 However,
the addition of polymers can reinforce this otherwise unstable
stream of colloids propagating in the centerline, since they act
as a buffer impeding the cross-stream migration that would
otherwise draw the colloids to the edges of the channel. The
rigid colloids do not suffer from any intrinsic penalty by existing in the centerline owing to their lack of internal degrees
of freedom. Instead, they are subjected solely to viscous drag
due to the no-slip boundary conditions at their surfaces. This
friction can be reduced for colloids that propagate in the wake
of others, promoting the emergence of well-ordered layers.
Concomitantly, the polymers are exposed to higher local shear
rates by migrating closer to the walls, resulting in a stronger
alignment in flow direction. This in turn leads to an overall
reduction of the intermolecular friction, since these off-center
planes of aligned polymers simply slip past one another as
opposed to tumbling in the channel center (see Fig. 8).
However, this balance cannot be sustained for arbitrarily
high flow rates as viscous forces become progressively less
significant as the dynamics are dominated by inertial effects
instead. Hence, the inertial lift forces experienced by the colloids become stronger with increasing flow strength, eventually allowing them to break through the polymer layers and
migrate to the channel edges. In order to quantify this be-
1
Np = 40
Np = 20
Np = 10
Np = 5
0.8
/ c
2
0.6
ctr
c
4
havior, we measured the fraction of colloids in the centerline,
ctr
c , as a function of Re. Here, we assign a colloid i to the
centerline when c (xi ) > 0.1 c (L/2) at its lateral position xi .
Although we do not a priori impose that this be symmetric,
we find that this occurs naturally to within a few per cent.
Figure 4(a) shows ctr
c for all investigated values of Np ,
and it is clearly visible that the range of Re over which colloids can be trapped in the centerline increases with Np . The
reason for this is twofold; first of all, the shear-alignment
of long polymers requires considerably lower γ̇ compared
to short ones, and therefore the stabilizing polymer buffer
emerges at lower Re. Second, it is more difficult for a colloid to migrate through a layer of long polymers as opposed
to short ones, since a layer of short shear-aligned polymers
contains less bonds on average than a layer of long polymers
with the equivalent monomer density. Thus, for short chains
there exist more gaps between them through which a colloid
can migrate while causing less significant perturbations to the
0.4
0.2
(a)
0
16
32
48
64
80
96
112
128
Re
1
c=
c=
c=
c=
0.8
/ c
Re = 16
Re = 48
Re = 112
0.075,
0.075,
0.075,
0.025,
m=
m=
m=
m=
0.025
0.050
0.075
0.050
0.6
ctr
c
c(x) c
3
6
J. Chem. Phys. 140, 094903 (2014)
0.4
0.2
(b)
0
16
32
48
64
80
96
112
128
Re
60
Rec
094903-5
40
20
Simulation data
Logarithmic curve fit
(c)
0
0
200
400
600
800
1000
1200
1400
p
FIG. 4. Panel (a): Fraction of colloids located in the channel center ctr
c vs.
Re for all investigated values of Np at fixed c = 0.075 and m = 0.050.
The symbols show our simulation data, while the continuous lines are fits
to a modified error function. Panel (b): Same as (a) but for all investigated
values of c and m at fixed Np = 40. Panel (c): Logarithmic curve fit of
Rec as a function of τ p for c = 0.075 and m = 0.050.
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Nikoubashman et al.
J. Chem. Phys. 140, 094903 (2014)
vL ∝
τp σc2 ∇ γ̇ (x)2 .
(9)
Equation (9) neglects the effect of the colloid diameter
σ c , which could influence the efficiency of focusing as well.
On the one hand, Eq. (8) suggests viscous lift forces ∝ σc3
toward the channel center in the regime Re 1. On the other
hand, it is well known from experiments46 and simulations16
of colloids in Newtonian solvents that the inertial lift forces
which push the colloids away from the centerline are proportional to σc3 in the regime Re 1. Hence, these two effects
are competing against each other at intermediate Re, and need
to be balanced in order to achieve an optimum focusing efficiency. To this end, we conducted additional simulations for
σ c = 6 and σ c = 10. Figure 5 shows ctr
c vs. Re, and it is clear
c= 6
c= 8
c = 10
/ c
0.8
0.6
0.4
(8)
This lift velocity vL has been derived in the limit of Re 1
and Pe 1, where inertial forces and thermal fluctuations can
be neglected entirely. Equation (8) suggests that all colloids
will eventually migrate to the center and remain there. This is
obviously not the case for nanoscale particles, since Brownian
motion can override vL and therefore drive particles out of the
centerline.
Figure 4(b) shows ctr
c for the investigated concentrations c and m , and it is readily apparent that increasing the
polymer concentration leads to a significant increase of both
ctr
c and Rec . Clearly, the overall reduction of viscous drag
achieved by replacing polymers with colloids in the centerline increases with m , until the centerline is entirely packed.
Moreover, it also becomes harder for colloids to drift away
from the centerline due to the denser polymer buffer layer.
This interpretation of the data is corroborated by the simulation results with lower c = 0.025, where ctr
c decays slightly
faster compared to the runs at c = 0.075, which might seem
counterintuitive at first glance. In this regard, colloids provide
a considerable amount of very mobile excluded volume which
allows the polymer to evacuate unfavorable channel regions,
whereupon taking the polymer’s place a colloid incurs no additional hydrodynamic penalties owing to its lack of internal
degrees of freedom.
From these measurements, we can suggest an empirical expression for Rec depending on the polymer relaxation
time τ p [via Np and Eq. (7)] and polymer concentration m .
To this end, we define Rec as the position of the inflection
55
point in ctr
c (Re), and determine its value by fitting our data
through a modified error function. Following our previous arguments, we can identify the two limiting cases as Rec → 0
for Np → 0 and Rec → ∞ for Np → ∞. The expression ln (1
+ τ p ) satisfies these two conditions, and fits our data well [see
Fig. 4(c)]. Furthermore, we found that Rec ∝ m for the m
values investigated here. Hence, we posit the following empirical relationship for Rec :
Rec ∝ m ln(1 + τp ).
1
ctr
c
otherwise flow-aligned polymers. Such perturbations would
incur a significant drag penalty, and impede such motion.
These findings are consistent with results from microfluidic
experiments on micrometer-sized colloids30, 31 and continuum
fluid dynamics calculations,53, 54 where an accelerated particle
motion toward the centerline was observed with increasing τ p
and γ̇
0.2
0
16
32
48
64
80
96
112
128
Re
FIG. 5. Fraction of colloids located in the channel center ctr
c vs. Re for all
investigated values of σ c at c = 0.075, m = 0.050, and Np = 40.
that ctr
c decays slower for larger solute particles. We also
observe a slight linear increase of Rec . This discrepancy stems
from the fact that the predictions for Re 1 are neglecting
viscoelastic effects, which still play a significant role even at
high flow rates.
The microscopic nature of our simulations allows us
to study the colloid trajectories and polymer conformations
in detail, which provide valuable additional insights into
the origin of the flow-induced demixing. In order to better
understand the breakup mechanism at Re > Rec , we conducted additional simulations in which we initialized the system from a focused state [cf. Fig. 2(e)], and tracked the lateral
position of individual colloids as a function of time. Figure 6
shows the trajectory of one such representative particle at intermediate and high Reynolds numbers. For Re = 48 < Rec ,
we can see that the colloid remains on its initial central position over the whole course of the simulation. As we increased
Re beyond its critical value, the particle abruptly left its centerline position and migrated toward the channel walls until it
reached a distance of Lx /4, and remained there for ≈4000 τ .
Subsequently, the colloid continued its outward motion until it eventually reached its steady state position close to the
wall. A similar transition can be observed for the highest flow
1
Re=48
Re=96
Re=128
0.75
x / Lx
094903-6
0.5
0.25
0
0
5000
10000
15000
20000
25000
t/
FIG. 6. Lateral colloid position x as a function of time t for c = 0.075
and m = 0.050. The system has been initialized from a focused configuration. Gray solid lines indicate layers where colloids aggregate and spend most
of their time. Separating these are layers of high polymer density as drawn
schematically in Fig. 2, which act as barriers impeding the movement of the
colloids toward the channel walls. These are progressively less significant as
Re increases.
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Nikoubashman et al.
094903-7
J. Chem. Phys. 140, 094903 (2014)
strengths investigated here (Re = 128). These data clearly
show the colloid propagating through well-defined layers of
polymers. At low Re, the colloids remain focused in the
centerline, however, upon increasing the flow strength they
migrate through the layers with increasing ease.
Next, we studied the polymer conformation at different
colloid concentrations and flow strengths. Figure 7(a) shows
the spatial distribution of the diagonal components of the gyαα
,
ration tensor normalized by its value at rest, Rgαα (x)/Rg,0
between the confining walls at Re = 16 for a polymer with
Np = 40 beads. We also measured the bond stretching due
to the applied flow, and found that the stretching was below
1% for the highest flow strengths applied. It is clear that the
polymer elongates in the flow direction and accordingly contracts in the remaining ones, though the reduction is always
more severe in the gradient direction owing to confinement.
However, this trend is significantly reduced very near the center of the channel due to the vanishing shear rate there; instead of forming a shear-aligned filament, the polymer follows
the parabolic velocity profile of the solvent and resembles a
“hairpin.” We observed the same qualitative behavior for all
investigated values of Np . Furthermore, it is clear that these
trends are independent of colloid concentration, which conforms with previous simulation results for single polymers in
solutions of colloids under shear flow.56
Figure 7(b) shows the normalized Rgαα as a function of
Re and Wi, where each data point has been obtained by
4
c =0.0,
c =0.025,
c =0.075,
(a)
3.5
2.5
2
1.5
1
0.5
0
0
0.25
0
3.5
35
0.5
x / Lx
0.75
<Wi>
105
140
70
1
175
2Rgxz
mg
= tan(2χ ) = zz
.
Wi
Rg − Rgxx
210
0.25
(b)
10
0.2
c=0.0,
c=0.025,
c=0.075,
2.5
2
m =0.050
m =0.050
m =0.050
0.15
1
0.1
1.5
/
(Re) / Rg,0
3
Rg
(10)
Figure 8 shows χ for the cases c = 0 and c = 0.075 at
Re = 96 as a function of x, and it is clear that, except near
the centerline, the polymers are nearly perfectly aligned with
the flow. Far from the centerline, complete shear-alignment
already took place at very low Re ≈ 16 and no significant
improvement was achieved upon increasing the flow strength
(not shown here). However, in the center of the channel,
where γ̇ = 0, polymer alignment is reduced dramatically and
tumbling leads to large variances in χ . This effect is amplified
as Re → 0 whereupon lower shear rates weaken the coupling
between the flow and polymer orientation. Tumbling polymers collide with their shear-aligned neighbors and thus increase their intermolecular friction; while this is unfavorable,
it is inevitable given the polymer density in the channel. This
trend is quantified in the inset of Fig. 8, where we plotted
mg as a function of Wi. It is well known from theory48 and
experiments57 that close to equilibrium, i.e., Wi → 0, the deformation of a sheared linear polymer is given by Rgxz ∝ γ̇
and (Rgzz − Rgxx ) ∝ γ̇ 2 . Hence, mG is independent of Wi in
this regime. For larger shear rates, the orientational resistance
mG
Rg
(x) / Rg,0
3
m=0.050
m=0.050
m=0.050
averaging over the channel width Lx . It is clearly visible that
the polymer chains are already almost fully aligned in the flow
direction at very low Re, and that the alignment quality does
not improve significantly for higher Re. This trend becomes
clear, considering that the corresponding Wi 1. Furthermore, Fig. 7(b) reveals that the average polymer extension in
the flow direction slightly increases for higher colloid concentrations c at intermediate Reynolds numbers Re < Rec .
This is due to the fact that more colloids occupy the centerline
and thereby push more polymers toward the channel edges,
where the local shear rate is higher. This effect vanishes for
Re > Rec , and Rgαα becomes independent of m .
We then quantified the average alignment of the polymers with respect to the flow by measuring orientational
resistance mg , based on the angle χ , between the eigenvector of the gyration tensor with the largest eigenvalue and
the flow direction.57, 58 For the system at hand, geometric
considerations yield
0.05
0.1
1
10
100
<Wi>
c=0.0,
c=0.075,
m =0.050
m =0.050
0
1
-0.05
0.5
-0.1
0
0
16
32
48
64
Re
80
96
112
128
FIG. 7. Panel (a): Diagonal components of the polymer gyration tensor Rgαα
at Re = 16, normalized by their values at rest, along the gradient direction
for a pure polymer system (lines) and mixtures (symbols). Panel (b): Rgαα
averaged over the channel width Lx as a function of Re and Wi. In both
panels, from top to bottom: flow (z), vorticity (y), and gradient direction (x).
-0.15
0
0.25
0.5
x / Lx
0.75
1
FIG. 8. Main plot: orientational resistance, mg , of polymers with Np in the
absence (red) and presence (blue) of colloids at Re = 96 (main). Inset: mG
from simulations as a function of Wi (symbols) together with its power law
fit (dashed line).
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094903-8
Nikoubashman et al.
can be described by a power law mG (Wi) ∝ Wiμ , where a
value of μ = 0.54 ± 0.03 has been reported in Ref. 59 for
ultradilute polymer solutions in Couette flow. We have measured a somewhat higher exponent μ = 0.76 ± 0.02, which
is likely due to the higher polymer concentration and inhomogeneous shear field in our simulations. From these data,
we can safely conclude that the presence of the spherical colloids does not have any significant impact on the shape of the
polymers for the parameters investigated.
IV. CONCLUSIONS
We performed extensive molecular dynamics simulations, coupled with Multi-Particle Collision Dynamics to take
into account solvent-mediated hydrodynamics, to study the
viscoelastic focusing of nanoparticles in microfluidic devices.
We explicitly modeled the constituent polymers of the viscoelastic medium and provided a detailed microscopic explanation of the phenomenon. We found that the emergence of
the steady stream of colloids in the centerline at low and intermediate Reynolds numbers Re is due to the cooperative minimization of viscous drag. Commensurately, as Re exceeded
some critical value Rec , inertial influences overrode this effect
leading to an abrupt decay in the centerline density peak. The
specific value of Rec strongly depends on the properties of
the viscoelastic medium, and we found that the colloid focusing could be maintained over a broader Re range for longer
polymers and higher polymer concentrations.
Although our investigation have provided a clearer picture of the mechanisms behind viscoelastic focusing of
nanoparticles, there still remain some open questions, such as
the influence of enthalpic polymer-colloid interactions. Furthermore, it is interesting to investigate the viscoelastic focusing of flexible and rigid polymers instead of rigid colloids,
since such scenarios are highly relevant for many biomedical
applications.
ACKNOWLEDGMENTS
A.N. acknowledges financial support from the Princeton Center for Complex Materials (PCCM), a U.S. National
Science Foundation (NSF) Materials Research Science and
Engineering Center (Grant No. DMR-0819860). In addition,
N.A.M. received support from the U.S. National Science
Foundation (Grant No. CBET-1033155).
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