Snell’s Law for Swimmers
Tyler D. Ross
1,*
, Dino Osmanović
2
, John F. Brady
3
, and Paul W. K. Rothemund
1
1
Department of Computing and Mathematical Sciences, California Institute of
Technology, Pasadena, CA 91125, USA.
2
Center for the Physics of Living Systems, Department of Physics, Massachusetts
Institute of Technology, Cambridge, Massachusetts 02139, USA
3
Divisions of Chemistry & Chemical Engineering and Engineering & Applied
Science, California Institute of Technology, Pasadena, CA 91125, USA
*
correspondence to: tross@caltech.edu
Snell’s law, which encompasses both refraction and total internal reflection, provides the
foundation for ray optics and all lens-based instruments, from microscopes to telescopes. Re-
fraction results when light crosses the interface between media of different refractive index, the
dimensionless number that captures how much a medium retards the propagation of light. In
this work, we show that the motion of self-propelled particles moving across a drag discontinuity
is governed by an analogous Snell’s law, allowing for swimmer ray optics. We derive a variant
of Snell’s law for swimmers moving across media of different viscosities. Just as the ratio of
refractive indexes sets the path of a light ray, the ratio of viscosities is shown to determine
the trajectories of swimmers. We find that the magnitude of refraction depends on the swim-
mer’s shape, specifically the aspect ratio, as analogous to the wavelength of light. This enables
the demixing of a polymorphic, many-shaped, beam of swimmers into distinct monomorphic,
single-shaped, beams through a viscosity prism. In turn, beams of monomorphic swimmers
can be focused by spherical and gradient viscosity lenses. Completing the analogy, we show
that the shape-dependence of the total internal reflection critical angle can be used to create
swimmer traps. Such analogies to ray optics suggest a universe of new devices for sorting,
concentrating, and analyzing microscopic swimmers is possible.
Introduction
Directed transport of matter on the microscale is a fundamental process in biology [1–3], lab-on-a-
chip systems [4, 5], and drug delivery [6–8]. One of the primary modes of directed transport relies
on the dynamics of micro-swimmers, or self-propelled particles. A substantial body of work has
demonstrated that swimmer density can be spatially modulated using asymmetric barriers [9, 10],
differing regions of propulsion speed [11–14], run-and-tumble dynamics [15], motility induced phase
separation [16], and dynamic swimmer affinities [17]. However, there are few known mechanisms
for controlling the trajectories of swimmers. Recently, there has been evidence that swimmers
refract and scatter at drag discontinuities [18–20]. While there is theory that studies the move-
ment of spherical swimmers down viscosity gradients [21] and theory that predicts that nonuniaxial
1
arXiv:2109.06360v3 [cond-mat.soft] 20 Sep 2021
swimmers can swim up viscosity gradients [22], there is no theory for swimmers encountering drag
discontinuities. Here we derive a Snell’s law for swimmers, which predicts how a swimmer’s tra-
jectory is changed when traversing a viscosity discontinuity and can be used to rationally design
optics-like devices for swimmers.
Derivation
We consider the trajectory of a swimmer moving across a drag discontinuity at low Reynold’s
number; for simplicity, we ignore hydrodynamic and Brownian interactions and assume that the
interface at the discontinuity does not deform as the swimmer passes through.
a
b
Constant
propulsive force
Reorientation
d
c
Reorientation
Figure 1: Sketch of spherical and rectangular swimmers moving across a drag (e.g. viscosity or
resistivity) discontinuity.
a
, Depiction of the relevant forces and torques on a swimmer as it travels
across a drag discontinuity.
b
, The forces and torques from (
a
) cause a reorientation of the swim-
mer’s trajectory around the drag discontinuity.
c
,
d
, Forces and re-orientation of a rectangular
swimmer depend on the aspect ratio of the swimmer.
As illustrated in Fig. 1, the non-accelerating motion of the swimmer follows from a balance
between the drag force,
F
drag
=
−
R
FU
·
U
, and a propulsive or swim force,
F
swim
:
0 =
−
R
FU
·
U
+
F
swim
.
(1)
In Eq. (1),
U
is the translational velocity of the swimmer, and
R
FU
is the (hydrodynamic) resistance
function that gives the coupling between the drag and the velocity. The resistance tensor depends
2
on the jump in resistivity, and the geometry—the size and shape of the swimmer as well as its
proximity and orientation relative to the discontinuity.
A swimmer is an active particle, pushing off its surroundings and generating a propulsive swim
force of the form:
F
swim
=
F
0
q
, where
F
0
is the magnitude and
q
the direction of propulsion. To
simplify the analysis, we take the resistance tensor to be isotropic and constant,
R
FU
=
ζ
tt
I
, where
ζ
tt
is the translational drag coefficient and
I
is the identity tensor. Thus, the velocity is
U
=
F
0
q
/ζ
tt
.
(2)
We consider a constant magnitude swim force
F
0
and thus the speed of the swimmer is slower in
the region of greater resistivity (or greater viscosity). For example, for a spherical swimmer in a
Newtonian fluid of viscosity
η
we have the well-known Stokes drag coefficient
ζ
tt
= 6
πηa
, where
a
is the swimmer radius.
The swimming direction
q
changes as a function of time according to
d
q
dt
=
Ω
×
q
,
(3)
where
Ω
is the angular velocity of the swimmer. The swimmer angular velocity follows from the
torque balance for the force- and torque-free motion
0 =
−
R
L
Ω
·
Ω
−
R
LU
·
U
.
(4)
In Eq. (4)
R
L
Ω
is the resistance tensor coupling the torque (
L
) to the angular velocity, and
R
LU
couples the torque to the translational velocity. As for the force-velocity coupling (
FU
), we take the
torque-angular velocity coupling to be isotropic and constant:
R
L
Ω
=
ζ
rr
I
; for a spherical swimmer
in a viscous fluid
ζ
rr
= 8
πηa
3
.
The torque-translational velocity (
LU
) coupling arises because as the swimmer crosses into a
region of higher resistivity that portion of the swimmer in the more resistive medium slows down
and thus the swimmer rotates such that its direction of motion tends to align along the normal
as illustrated in Fig. 1a. The opposite occurs when moving into a less resistive medium. The
LU
coupling is a pseudo tensor and since the swimmer itself is not chiral, it must be of the form
R
LU
=
ζ
rt
·
n
, where
is the unit alternating tensor,
n
is the normal to the discontinuity, and
ζ
rt
is the drag coefficient. For a sphere, the
LU
coupling only arises if there is a jump in resistivity,
ζ
rt
∼
6
π
∆
ηa
2
.
Combining Eqs. (1) and (4), Eq. (3) becomes
d
q
dt
=
ζ
rt
ζ
rr
F
0
ζ
tt
(
n
×
q
)
×
q
=
ζ
rt
ζ
rr
F
0
ζ
tt
[
n
−
q
(
q
·
n
)]
.
(5)
Now,
n
·
q
= cos
θ
, where
θ
is the angle between the normal and the swim direction, and thus
Eq. (5) gives an equation for the evolution of
θ
(
t
)
:
d
cos
θ
sin
2
θ
=
ζ
rt
ζ
rr
F
0
ζ
tt
dt.
(6)
We need to integrate Eq. (6) from the time the swimmer first touches the discontinuity (
t
= 0
)
with incident angle
θ
0
until it fully crosses into the next region at the final time
t
f
, which will then
give the out-going angle
θ
f
. The time to cross the interface follows from the translational velocity
3
d
x
/dt
=
U
, and since only the normal component of the velocity is responsible for the swimmer
crossing we have
d
(
n
·
x
)
dt
=
F
0
ζ
tt
n
·
q
=
F
0
ζ
tt
cos
θ .
(7)
We can use Eq. (7) to replace
dt
in Eq. (6) to give
dx
⊥
=
ζ
rr
ζ
rt
cos
θ d
cos
θ
sin
2
θ
=
−
ζ
rr
ζ
rt
d
ln(sin
θ
)
,
(8)
where
x
⊥
=
n
·
x
is the amount of the swimmer that has crossed the interface. For a spherical
swimmer, integrating from 0 to
2
a
relates the initial to the final angle and yields a Snell’s law:
sin
θ
f
=
e
α
sin
θ
0
,
(9)
where
α
=
−
2
aζ
rt
/ζ
rr
. This behavior is independent of the magnitude of the propulsive force
F
0
and the Stokes drag
ζ
tt
. Further by dimensional arguments, the drag coefficient for
LU
coupling is
proportional to
a
2
and thus
α
is also independent of the size of the swimmer. The validity of this
Snell’s law and its independence on the swimmer size are verified by direct simulation below.
We have made a number of approximations in arriving at this Snell’s law. First, we have assumed
that the drag coefficients
ζ
rr
and
ζ
rt
are constants (Note that
ζ
tt
cancels out in Eq. (8)). Both are
proportional to the local value of the resistance or viscosity of the medium and thus depend on the
portion of the swimmer in each region. We can include this effect by noting that Eq. (8) can be
written as
ζ
rt
(
x
⊥
)
ζ
rr
(
x
⊥
)
dx
⊥
=
−
d
ln(sin
θ
)
,
(10)
and integration from
0
to
2
a
again recovers Snell’s law Eq. (9) where
α
is now given by
α
=
−
∫
2
a
0
ζ
rt
(
x
⊥
)
ζ
rr
(
x
⊥
)
dx
⊥
.
(11)
From a micromechanical model for
ζ
rr
(
x
⊥
)
and
ζ
rt
(
x
⊥
)
one could compute
α
; the result will be the
form
α
=
−
C
∆
η
〈
η
〉
,
(12)
where
∆
η
=
η
f
−
η
0
,
〈
η
〉
= (
η
f
+
η
0
)
/
2
and
C
is an order 1 constant that is (weakly) dependent
of the viscosity ratio
η
f
/η
0
(Fig. 1b). Later we show that Snell’s law with Eq. (12) is in excellent
agreement with detailed micromechanical simulations.
Since the reorientation arises because part of the swimmer finds itself in a more resistive medium,
if the swimmer is very thin relative to its swim axis, then the differential resistance across the body
is small and the reorientation should be reduced. An infinitely thin swimmer will not reorient at
all. We can account for this shape effect in a simple manner by recognizing that the amount of
the swimmer that has crossed the discontinuity
∆
x
⊥
depends on the body shape and the initial
orientation
θ
0
. For a simple rectangular swimmer shown in Fig. 1c,d,
∆
x
⊥
=
`
cos
θ
0
+
a
(1
−
cos
θ
0
)
,
(13)
where
`
is the half major length and
a
is the half minor length. Using this in Eq. (8) we again have
Snell’s law, but now
α
∼−
a
`
2
(
a
+ (
`
−
a
) cos
θ
0
)
∆
η
〈
η
〉
,
(14)
4
where we have used the geometric scaling that
ζ
rt
/ζ
rr
∼
a/`
2
. We have also assumed the instanta-
neous orientation angle of the body could be approximated with its initial angle
θ
0
. When
`
=
a
,
Eq. (14) reduces to Eq. (12). The dependence on the aspect ratio is similar to the wavelength
dependence of the refractive index. We later use numerical simulations to test this prediction.
We mention a few caveats. Above we assumed that any force-angular velocity coupling
R
F
Ω
=
R
†
LU
was negligible. If
R
F
Ω
is included, the force balance Eq. (1) now becomes
0 =
−
R
FU
·
U
−
R
F
Ω
·
Ω
+
F
swim
, and, when combined with the angular momentum balance Eq. (4), will give an angular
velocity
Ω
=
−
R
−
1
L
Ω
·
R
LU
·
[
R
FU
−
R
F
Ω
·
R
−
1
L
Ω
·
R
LU
]
−
1
·
F
swim
. The additional factor
R
F
Ω
·
R
−
1
L
Ω
·
R
LU
will now add an additional
θ
dependence to Eq. (6) and we can only derive a Snell’s law under
the condition that
ζ
2
rt
/
(
ζ
rr
ζ
tt
)
1
. Furthermore non-spherical swimmers will not, in general, have
isotropic resistance tensors and we can expect, e.g., the force-velocity coupling to have the form
R
FU
=
ζ
‖
qq
+
ζ
⊥
(
I
−
qq
)
, where
ζ
‖
and
ζ
⊥
are drag coefficients for motion parallel and perpendicular
to the swimmer axis, which we assume be the same as the direction of propulsion
q
. There will also
be similar forms for
R
L
Ω
and
R
LU
. Clearly, this complicates the analysis and in general it may be
impossible to find a Snell-like analytical expression for the refraction of nonspherical swimmers.
Finally, our neglect of hydrodynamic interactions enables us to assume that there is no interac-
tion between the swimmer and the drag discontinuity until they make physical contact. In general,
the introduction of hydrodynamic interactions will make resistance tensors depend on the distance
of the swimmer from the discontinuity. The magnitude of such effects will depend on the swimmer
type—pusher, puller, or neutral—and the magnitude of the discontinuity. Neutral swimmers, with
their more uniform flow fields, might be expected to give refraction similar to that predicted here,
but in general but it is unlikely that simple Snell’s law will emerge once hydrodynamics is con-
sidered. Two observations are worth making. First, the force- and torque-balance approach taken
here will apply and contribute terms to any full hydrodynamic treatment which seeks to calculate
the refraction of a swimmer moving through a drag discontinuity. Second, for crawlers [23] whose
propulsive force is generated by interaction with a surface rather than a fluid, the Snell’s law derived
here may apply with little or no modification.
Analysis of results
Simulations of the spherical swimmer closely agree with Eqs. (9) and (12) (Fig. 2a, see Methods and
Supplementary Information for detail of simulation and curve fitting; see Video 1 for an example
simulation). One prediction from our theory is that refraction is size independent. By simulating
swimmers of different sizes, we find that the angle of refraction is indeed size invariant (Fig. S2a).
Further, the form of our Snell’s law for a spherical swimmer implies that there is a symmetry about
the line
θ
f
=
θ
0
. We are able to verify this prediction by comparing the data points and curves
across this axis of symmetry (Fig. S2b).
As for Snell’s law, when
η
f
/η
0
<
1
, Eq. (9) is valid up to
θ
f
=
π
2
. For
θ
f
=
π
2
, the incident
critical angle is
θ
crit
= arcsin
e
−
α
.
(15)
For
θ
0
> θ
crit
swimmers obey the law of reflection,
θ
f
=
π
−
θ
0
,
(16)
which we confirm with simulations (Fig. 2b). Through symmetry, the critical angle is the same as
Snell’s window, which is the greatest possible refraction angle for a given
η
f
/η
0
>
1
.
5